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Properties

Label 20-2175e10-1.1-c3e10-0-0
Degree 2020
Conductor 2.369×10332.369\times 10^{33}
Sign 11
Analytic cond. 1.21130×10211.21130\times 10^{21}
Root an. cond. 11.328211.3282
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 30·3-s + 4-s + 120·6-s − 75·7-s + 25·8-s + 495·9-s + 3·11-s − 30·12-s − 75·13-s + 300·14-s − 85·16-s − 131·17-s − 1.98e3·18-s + 264·19-s + 2.25e3·21-s − 12·22-s − 204·23-s − 750·24-s + 300·26-s − 5.94e3·27-s − 75·28-s − 290·29-s + 924·31-s + 183·32-s − 90·33-s + 524·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 5.77·3-s + 1/8·4-s + 8.16·6-s − 4.04·7-s + 1.10·8-s + 55/3·9-s + 0.0822·11-s − 0.721·12-s − 1.60·13-s + 5.72·14-s − 1.32·16-s − 1.86·17-s − 25.9·18-s + 3.18·19-s + 23.3·21-s − 0.116·22-s − 1.84·23-s − 6.37·24-s + 2.26·26-s − 42.3·27-s − 0.506·28-s − 1.85·29-s + 5.35·31-s + 1.01·32-s − 0.474·33-s + 2.64·34-s + ⋯

Functional equation

Λ(s)=((310⋅520⋅2910)s/2 ΓC(s)10 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((310⋅520⋅2910)s/2 ΓC(s+3/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 310⋅520⋅29103^{10} \cdot 5^{20} \cdot 29^{10}
Sign: 11
Analytic conductor: 1.21130×10211.21130\times 10^{21}
Root analytic conductor: 11.328211.3282
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 310⋅520⋅2910, ( :[3/2]10), 1)(20,\ 3^{10} \cdot 5^{20} \cdot 29^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.18522764030.1852276403
L(12)L(\frac12) ≈\approx 0.18522764030.1852276403
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 (1+pT)10 ( 1 + p T )^{10}
5 1 1
29 (1+pT)10 ( 1 + p T )^{10}
good2 1+p2T+15T2+31T3+47pT4+127T5+175pT6−269pT7−81p2T8−239p4T9+367p6T10−239p7T11−81p8T12−269p10T13+175p13T14+127p15T15+47p19T16+31p21T17+15p24T18+p29T19+p30T20 1 + p^{2} T + 15 T^{2} + 31 T^{3} + 47 p T^{4} + 127 T^{5} + 175 p T^{6} - 269 p T^{7} - 81 p^{2} T^{8} - 239 p^{4} T^{9} + 367 p^{6} T^{10} - 239 p^{7} T^{11} - 81 p^{8} T^{12} - 269 p^{10} T^{13} + 175 p^{13} T^{14} + 127 p^{15} T^{15} + 47 p^{19} T^{16} + 31 p^{21} T^{17} + 15 p^{24} T^{18} + p^{29} T^{19} + p^{30} T^{20}
7 1+75T+4349T2+181642T3+6656673T4+206630229T5+5833261716T6+146138007193T7+3379984441646T8+70697748137451T9+196202793405810pT10+70697748137451p3T11+3379984441646p6T12+146138007193p9T13+5833261716p12T14+206630229p15T15+6656673p18T16+181642p21T17+4349p24T18+75p27T19+p30T20 1 + 75 T + 4349 T^{2} + 181642 T^{3} + 6656673 T^{4} + 206630229 T^{5} + 5833261716 T^{6} + 146138007193 T^{7} + 3379984441646 T^{8} + 70697748137451 T^{9} + 196202793405810 p T^{10} + 70697748137451 p^{3} T^{11} + 3379984441646 p^{6} T^{12} + 146138007193 p^{9} T^{13} + 5833261716 p^{12} T^{14} + 206630229 p^{15} T^{15} + 6656673 p^{18} T^{16} + 181642 p^{21} T^{17} + 4349 p^{24} T^{18} + 75 p^{27} T^{19} + p^{30} T^{20}
11 1−3T+6228T2−33867T3+19457362T4−193113981T5+43727711526T6−548413433115T7+79124653287245T8−958830278761134T9+116734800107119276T10−958830278761134p3T11+79124653287245p6T12−548413433115p9T13+43727711526p12T14−193113981p15T15+19457362p18T16−33867p21T17+6228p24T18−3p27T19+p30T20 1 - 3 T + 6228 T^{2} - 33867 T^{3} + 19457362 T^{4} - 193113981 T^{5} + 43727711526 T^{6} - 548413433115 T^{7} + 79124653287245 T^{8} - 958830278761134 T^{9} + 116734800107119276 T^{10} - 958830278761134 p^{3} T^{11} + 79124653287245 p^{6} T^{12} - 548413433115 p^{9} T^{13} + 43727711526 p^{12} T^{14} - 193113981 p^{15} T^{15} + 19457362 p^{18} T^{16} - 33867 p^{21} T^{17} + 6228 p^{24} T^{18} - 3 p^{27} T^{19} + p^{30} T^{20}
13 1+75T+10297T2+724904T3+59406937T4+3736300879T5+239448338036T6+13847883384393T7+741713663817214T8+39022215828823561T9+140715758908462622pT10+39022215828823561p3T11+741713663817214p6T12+13847883384393p9T13+239448338036p12T14+3736300879p15T15+59406937p18T16+724904p21T17+10297p24T18+75p27T19+p30T20 1 + 75 T + 10297 T^{2} + 724904 T^{3} + 59406937 T^{4} + 3736300879 T^{5} + 239448338036 T^{6} + 13847883384393 T^{7} + 741713663817214 T^{8} + 39022215828823561 T^{9} + 140715758908462622 p T^{10} + 39022215828823561 p^{3} T^{11} + 741713663817214 p^{6} T^{12} + 13847883384393 p^{9} T^{13} + 239448338036 p^{12} T^{14} + 3736300879 p^{15} T^{15} + 59406937 p^{18} T^{16} + 724904 p^{21} T^{17} + 10297 p^{24} T^{18} + 75 p^{27} T^{19} + p^{30} T^{20}
17 1+131T+18851T2+2063592T3+186346487T4+14919655843T5+1209057672744T6+76744635197001T7+5831013350958248T8+387146061111209753T9+26351653051154600938T10+387146061111209753p3T11+5831013350958248p6T12+76744635197001p9T13+1209057672744p12T14+14919655843p15T15+186346487p18T16+2063592p21T17+18851p24T18+131p27T19+p30T20 1 + 131 T + 18851 T^{2} + 2063592 T^{3} + 186346487 T^{4} + 14919655843 T^{5} + 1209057672744 T^{6} + 76744635197001 T^{7} + 5831013350958248 T^{8} + 387146061111209753 T^{9} + 26351653051154600938 T^{10} + 387146061111209753 p^{3} T^{11} + 5831013350958248 p^{6} T^{12} + 76744635197001 p^{9} T^{13} + 1209057672744 p^{12} T^{14} + 14919655843 p^{15} T^{15} + 186346487 p^{18} T^{16} + 2063592 p^{21} T^{17} + 18851 p^{24} T^{18} + 131 p^{27} T^{19} + p^{30} T^{20}
19 1−264T+49690T2−5884480T3+646999973T4−57535351432T5+5638274075320T6−453767024009096T7+40345751365807186T8−2879054647357398936T9+ 1 - 264 T + 49690 T^{2} - 5884480 T^{3} + 646999973 T^{4} - 57535351432 T^{5} + 5638274075320 T^{6} - 453767024009096 T^{7} + 40345751365807186 T^{8} - 2879054647357398936 T^{9} + 25 ⁣⋯ ⁣6025\!\cdots\!60T10−2879054647357398936p3T11+40345751365807186p6T12−453767024009096p9T13+5638274075320p12T14−57535351432p15T15+646999973p18T16−5884480p21T17+49690p24T18−264p27T19+p30T20 T^{10} - 2879054647357398936 p^{3} T^{11} + 40345751365807186 p^{6} T^{12} - 453767024009096 p^{9} T^{13} + 5638274075320 p^{12} T^{14} - 57535351432 p^{15} T^{15} + 646999973 p^{18} T^{16} - 5884480 p^{21} T^{17} + 49690 p^{24} T^{18} - 264 p^{27} T^{19} + p^{30} T^{20}
23 1+204T+85735T2+13442446T3+3227661233T4+432238552034T5+77287479740476T6+9377158932056890T7+1374659627694043454T8+ 1 + 204 T + 85735 T^{2} + 13442446 T^{3} + 3227661233 T^{4} + 432238552034 T^{5} + 77287479740476 T^{6} + 9377158932056890 T^{7} + 1374659627694043454 T^{8} + 15 ⁣⋯ ⁣6215\!\cdots\!62T9+ T^{9} + 18 ⁣⋯ ⁣9418\!\cdots\!94T10+ T^{10} + 15 ⁣⋯ ⁣6215\!\cdots\!62p3T11+1374659627694043454p6T12+9377158932056890p9T13+77287479740476p12T14+432238552034p15T15+3227661233p18T16+13442446p21T17+85735p24T18+204p27T19+p30T20 p^{3} T^{11} + 1374659627694043454 p^{6} T^{12} + 9377158932056890 p^{9} T^{13} + 77287479740476 p^{12} T^{14} + 432238552034 p^{15} T^{15} + 3227661233 p^{18} T^{16} + 13442446 p^{21} T^{17} + 85735 p^{24} T^{18} + 204 p^{27} T^{19} + p^{30} T^{20}
31 1−924T+481922T2−171484788T3+47268473485T4−10848035817696T5+2262948355557848T6−455712768158393344T7+2935429135812927246pT8− 1 - 924 T + 481922 T^{2} - 171484788 T^{3} + 47268473485 T^{4} - 10848035817696 T^{5} + 2262948355557848 T^{6} - 455712768158393344 T^{7} + 2935429135812927246 p T^{8} - 17 ⁣⋯ ⁣0817\!\cdots\!08T9+ T^{9} + 31 ⁣⋯ ⁣3631\!\cdots\!36T10− T^{10} - 17 ⁣⋯ ⁣0817\!\cdots\!08p3T11+2935429135812927246p7T12−455712768158393344p9T13+2262948355557848p12T14−10848035817696p15T15+47268473485p18T16−171484788p21T17+481922p24T18−924p27T19+p30T20 p^{3} T^{11} + 2935429135812927246 p^{7} T^{12} - 455712768158393344 p^{9} T^{13} + 2262948355557848 p^{12} T^{14} - 10848035817696 p^{15} T^{15} + 47268473485 p^{18} T^{16} - 171484788 p^{21} T^{17} + 481922 p^{24} T^{18} - 924 p^{27} T^{19} + p^{30} T^{20}
37 1+212T+244875T2+63865934T3+34071945753T4+9438249194838T5+3290715983025020T6+911944517057300106T7+ 1 + 212 T + 244875 T^{2} + 63865934 T^{3} + 34071945753 T^{4} + 9438249194838 T^{5} + 3290715983025020 T^{6} + 911944517057300106 T^{7} + 23 ⁣⋯ ⁣2223\!\cdots\!22T8+ T^{8} + 62 ⁣⋯ ⁣2662\!\cdots\!26T9+ T^{9} + 13 ⁣⋯ ⁣0613\!\cdots\!06T10+ T^{10} + 62 ⁣⋯ ⁣2662\!\cdots\!26p3T11+ p^{3} T^{11} + 23 ⁣⋯ ⁣2223\!\cdots\!22p6T12+911944517057300106p9T13+3290715983025020p12T14+9438249194838p15T15+34071945753p18T16+63865934p21T17+244875p24T18+212p27T19+p30T20 p^{6} T^{12} + 911944517057300106 p^{9} T^{13} + 3290715983025020 p^{12} T^{14} + 9438249194838 p^{15} T^{15} + 34071945753 p^{18} T^{16} + 63865934 p^{21} T^{17} + 244875 p^{24} T^{18} + 212 p^{27} T^{19} + p^{30} T^{20}
41 1+192T+412143T2+60903586T3+86214319565T4+10490909875438T5+12061213397017476T6+1250206205177719290T7+ 1 + 192 T + 412143 T^{2} + 60903586 T^{3} + 86214319565 T^{4} + 10490909875438 T^{5} + 12061213397017476 T^{6} + 1250206205177719290 T^{7} + 12 ⁣⋯ ⁣2612\!\cdots\!26T8+ T^{8} + 11 ⁣⋯ ⁣0211\!\cdots\!02T9+ T^{9} + 97 ⁣⋯ ⁣2297\!\cdots\!22T10+ T^{10} + 11 ⁣⋯ ⁣0211\!\cdots\!02p3T11+ p^{3} T^{11} + 12 ⁣⋯ ⁣2612\!\cdots\!26p6T12+1250206205177719290p9T13+12061213397017476p12T14+10490909875438p15T15+86214319565p18T16+60903586p21T17+412143p24T18+192p27T19+p30T20 p^{6} T^{12} + 1250206205177719290 p^{9} T^{13} + 12061213397017476 p^{12} T^{14} + 10490909875438 p^{15} T^{15} + 86214319565 p^{18} T^{16} + 60903586 p^{21} T^{17} + 412143 p^{24} T^{18} + 192 p^{27} T^{19} + p^{30} T^{20}
43 1+614T+559999T2+256041822T3+145627636581T4+54262712047260T5+24016915481183380T6+7692373697421922076T7+ 1 + 614 T + 559999 T^{2} + 256041822 T^{3} + 145627636581 T^{4} + 54262712047260 T^{5} + 24016915481183380 T^{6} + 7692373697421922076 T^{7} + 28 ⁣⋯ ⁣4628\!\cdots\!46T8+ T^{8} + 80 ⁣⋯ ⁣9680\!\cdots\!96T9+ T^{9} + 25 ⁣⋯ ⁣7025\!\cdots\!70T10+ T^{10} + 80 ⁣⋯ ⁣9680\!\cdots\!96p3T11+ p^{3} T^{11} + 28 ⁣⋯ ⁣4628\!\cdots\!46p6T12+7692373697421922076p9T13+24016915481183380p12T14+54262712047260p15T15+145627636581p18T16+256041822p21T17+559999p24T18+614p27T19+p30T20 p^{6} T^{12} + 7692373697421922076 p^{9} T^{13} + 24016915481183380 p^{12} T^{14} + 54262712047260 p^{15} T^{15} + 145627636581 p^{18} T^{16} + 256041822 p^{21} T^{17} + 559999 p^{24} T^{18} + 614 p^{27} T^{19} + p^{30} T^{20}
47 1+173T+604281T2+77409878T3+170107620517T4+13663097450483T5+30107873803292316T6+1100188730698156631T7+ 1 + 173 T + 604281 T^{2} + 77409878 T^{3} + 170107620517 T^{4} + 13663097450483 T^{5} + 30107873803292316 T^{6} + 1100188730698156631 T^{7} + 39 ⁣⋯ ⁣9839\!\cdots\!98T8+ T^{8} + 30 ⁣⋯ ⁣2930\!\cdots\!29T9+ T^{9} + 43 ⁣⋯ ⁣9443\!\cdots\!94T10+ T^{10} + 30 ⁣⋯ ⁣2930\!\cdots\!29p3T11+ p^{3} T^{11} + 39 ⁣⋯ ⁣9839\!\cdots\!98p6T12+1100188730698156631p9T13+30107873803292316p12T14+13663097450483p15T15+170107620517p18T16+77409878p21T17+604281p24T18+173p27T19+p30T20 p^{6} T^{12} + 1100188730698156631 p^{9} T^{13} + 30107873803292316 p^{12} T^{14} + 13663097450483 p^{15} T^{15} + 170107620517 p^{18} T^{16} + 77409878 p^{21} T^{17} + 604281 p^{24} T^{18} + 173 p^{27} T^{19} + p^{30} T^{20}
53 1+658T+851043T2+438313758T3+333130613549T4+142613148718444T5+85246198893996068T6+32140168929964093684T7+ 1 + 658 T + 851043 T^{2} + 438313758 T^{3} + 333130613549 T^{4} + 142613148718444 T^{5} + 85246198893996068 T^{6} + 32140168929964093684 T^{7} + 16 ⁣⋯ ⁣1816\!\cdots\!18T8+ T^{8} + 57 ⁣⋯ ⁣5257\!\cdots\!52T9+ T^{9} + 27 ⁣⋯ ⁣1827\!\cdots\!18T10+ T^{10} + 57 ⁣⋯ ⁣5257\!\cdots\!52p3T11+ p^{3} T^{11} + 16 ⁣⋯ ⁣1816\!\cdots\!18p6T12+32140168929964093684p9T13+85246198893996068p12T14+142613148718444p15T15+333130613549p18T16+438313758p21T17+851043p24T18+658p27T19+p30T20 p^{6} T^{12} + 32140168929964093684 p^{9} T^{13} + 85246198893996068 p^{12} T^{14} + 142613148718444 p^{15} T^{15} + 333130613549 p^{18} T^{16} + 438313758 p^{21} T^{17} + 851043 p^{24} T^{18} + 658 p^{27} T^{19} + p^{30} T^{20}
59 1−190T+1162874T2−115602698T3+654603455477T4−18529287790584T5+244034807765660600T6+5572841685992684344T7+ 1 - 190 T + 1162874 T^{2} - 115602698 T^{3} + 654603455477 T^{4} - 18529287790584 T^{5} + 244034807765660600 T^{6} + 5572841685992684344 T^{7} + 68 ⁣⋯ ⁣4668\!\cdots\!46T8+ T^{8} + 33 ⁣⋯ ⁣4433\!\cdots\!44T9+ T^{9} + 15 ⁣⋯ ⁣8415\!\cdots\!84T10+ T^{10} + 33 ⁣⋯ ⁣4433\!\cdots\!44p3T11+ p^{3} T^{11} + 68 ⁣⋯ ⁣4668\!\cdots\!46p6T12+5572841685992684344p9T13+244034807765660600p12T14−18529287790584p15T15+654603455477p18T16−115602698p21T17+1162874p24T18−190p27T19+p30T20 p^{6} T^{12} + 5572841685992684344 p^{9} T^{13} + 244034807765660600 p^{12} T^{14} - 18529287790584 p^{15} T^{15} + 654603455477 p^{18} T^{16} - 115602698 p^{21} T^{17} + 1162874 p^{24} T^{18} - 190 p^{27} T^{19} + p^{30} T^{20}
61 1−1452T+2402718T2−2378615028T3+2381199022661T4−1829799967832888T5+1380956800645683048T6− 1 - 1452 T + 2402718 T^{2} - 2378615028 T^{3} + 2381199022661 T^{4} - 1829799967832888 T^{5} + 1380956800645683048 T^{6} - 86 ⁣⋯ ⁣4886\!\cdots\!48T7+ T^{7} + 53 ⁣⋯ ⁣2653\!\cdots\!26T8− T^{8} - 28 ⁣⋯ ⁣4428\!\cdots\!44T9+ T^{9} + 14 ⁣⋯ ⁣5214\!\cdots\!52T10− T^{10} - 28 ⁣⋯ ⁣4428\!\cdots\!44p3T11+ p^{3} T^{11} + 53 ⁣⋯ ⁣2653\!\cdots\!26p6T12− p^{6} T^{12} - 86 ⁣⋯ ⁣4886\!\cdots\!48p9T13+1380956800645683048p12T14−1829799967832888p15T15+2381199022661p18T16−2378615028p21T17+2402718p24T18−1452p27T19+p30T20 p^{9} T^{13} + 1380956800645683048 p^{12} T^{14} - 1829799967832888 p^{15} T^{15} + 2381199022661 p^{18} T^{16} - 2378615028 p^{21} T^{17} + 2402718 p^{24} T^{18} - 1452 p^{27} T^{19} + p^{30} T^{20}
67 1+343T+1564407T2+123013642T3+1036964472039T4−182771991337139T5+453294026885687224T6− 1 + 343 T + 1564407 T^{2} + 123013642 T^{3} + 1036964472039 T^{4} - 182771991337139 T^{5} + 453294026885687224 T^{6} - 16 ⁣⋯ ⁣5116\!\cdots\!51T7+ T^{7} + 17 ⁣⋯ ⁣0417\!\cdots\!04T8− T^{8} - 72 ⁣⋯ ⁣7372\!\cdots\!73T9+ T^{9} + 57 ⁣⋯ ⁣8657\!\cdots\!86T10− T^{10} - 72 ⁣⋯ ⁣7372\!\cdots\!73p3T11+ p^{3} T^{11} + 17 ⁣⋯ ⁣0417\!\cdots\!04p6T12− p^{6} T^{12} - 16 ⁣⋯ ⁣5116\!\cdots\!51p9T13+453294026885687224p12T14−182771991337139p15T15+1036964472039p18T16+123013642p21T17+1564407p24T18+343p27T19+p30T20 p^{9} T^{13} + 453294026885687224 p^{12} T^{14} - 182771991337139 p^{15} T^{15} + 1036964472039 p^{18} T^{16} + 123013642 p^{21} T^{17} + 1564407 p^{24} T^{18} + 343 p^{27} T^{19} + p^{30} T^{20}
71 1−8T+2054898T2+350475824T3+1940327221213T4+722485340128872T5+1144209103579187736T6+ 1 - 8 T + 2054898 T^{2} + 350475824 T^{3} + 1940327221213 T^{4} + 722485340128872 T^{5} + 1144209103579187736 T^{6} + 67 ⁣⋯ ⁣2067\!\cdots\!20T7+ T^{7} + 50 ⁣⋯ ⁣7450\!\cdots\!74T8+ T^{8} + 37 ⁣⋯ ⁣0037\!\cdots\!00T9+ T^{9} + 18 ⁣⋯ ⁣4018\!\cdots\!40T10+ T^{10} + 37 ⁣⋯ ⁣0037\!\cdots\!00p3T11+ p^{3} T^{11} + 50 ⁣⋯ ⁣7450\!\cdots\!74p6T12+ p^{6} T^{12} + 67 ⁣⋯ ⁣2067\!\cdots\!20p9T13+1144209103579187736p12T14+722485340128872p15T15+1940327221213p18T16+350475824p21T17+2054898p24T18−8p27T19+p30T20 p^{9} T^{13} + 1144209103579187736 p^{12} T^{14} + 722485340128872 p^{15} T^{15} + 1940327221213 p^{18} T^{16} + 350475824 p^{21} T^{17} + 2054898 p^{24} T^{18} - 8 p^{27} T^{19} + p^{30} T^{20}
73 1+1500T+2910491T2+3080902530T3+3767645887721T4+3256214674125362T5+3090987939219640012T6+ 1 + 1500 T + 2910491 T^{2} + 3080902530 T^{3} + 3767645887721 T^{4} + 3256214674125362 T^{5} + 3090987939219640012 T^{6} + 22 ⁣⋯ ⁣8222\!\cdots\!82T7+ T^{7} + 18 ⁣⋯ ⁣3418\!\cdots\!34T8+ T^{8} + 11 ⁣⋯ ⁣0611\!\cdots\!06T9+ T^{9} + 80 ⁣⋯ ⁣6280\!\cdots\!62T10+ T^{10} + 11 ⁣⋯ ⁣0611\!\cdots\!06p3T11+ p^{3} T^{11} + 18 ⁣⋯ ⁣3418\!\cdots\!34p6T12+ p^{6} T^{12} + 22 ⁣⋯ ⁣8222\!\cdots\!82p9T13+3090987939219640012p12T14+3256214674125362p15T15+3767645887721p18T16+3080902530p21T17+2910491p24T18+1500p27T19+p30T20 p^{9} T^{13} + 3090987939219640012 p^{12} T^{14} + 3256214674125362 p^{15} T^{15} + 3767645887721 p^{18} T^{16} + 3080902530 p^{21} T^{17} + 2910491 p^{24} T^{18} + 1500 p^{27} T^{19} + p^{30} T^{20}
79 1−2438T+5415286T2−7517245138T3+9411524773309T4−8926607546744448T5+7716663020937405928T6− 1 - 2438 T + 5415286 T^{2} - 7517245138 T^{3} + 9411524773309 T^{4} - 8926607546744448 T^{5} + 7716663020937405928 T^{6} - 53 ⁣⋯ ⁣7653\!\cdots\!76T7+ T^{7} + 35 ⁣⋯ ⁣9435\!\cdots\!94T8− T^{8} - 20 ⁣⋯ ⁣2420\!\cdots\!24T9+ T^{9} + 14 ⁣⋯ ⁣4414\!\cdots\!44T10− T^{10} - 20 ⁣⋯ ⁣2420\!\cdots\!24p3T11+ p^{3} T^{11} + 35 ⁣⋯ ⁣9435\!\cdots\!94p6T12− p^{6} T^{12} - 53 ⁣⋯ ⁣7653\!\cdots\!76p9T13+7716663020937405928p12T14−8926607546744448p15T15+9411524773309p18T16−7517245138p21T17+5415286p24T18−2438p27T19+p30T20 p^{9} T^{13} + 7716663020937405928 p^{12} T^{14} - 8926607546744448 p^{15} T^{15} + 9411524773309 p^{18} T^{16} - 7517245138 p^{21} T^{17} + 5415286 p^{24} T^{18} - 2438 p^{27} T^{19} + p^{30} T^{20}
83 1+1336T+3369527T2+2459621218T3+3945615288257T4+1456880186173970T5+2681144587556984476T6+ 1 + 1336 T + 3369527 T^{2} + 2459621218 T^{3} + 3945615288257 T^{4} + 1456880186173970 T^{5} + 2681144587556984476 T^{6} + 32 ⁣⋯ ⁣2632\!\cdots\!26T7+ T^{7} + 16 ⁣⋯ ⁣0216\!\cdots\!02T8+ T^{8} + 77 ⁣⋯ ⁣9077\!\cdots\!90T9+ T^{9} + 10 ⁣⋯ ⁣3010\!\cdots\!30T10+ T^{10} + 77 ⁣⋯ ⁣9077\!\cdots\!90p3T11+ p^{3} T^{11} + 16 ⁣⋯ ⁣0216\!\cdots\!02p6T12+ p^{6} T^{12} + 32 ⁣⋯ ⁣2632\!\cdots\!26p9T13+2681144587556984476p12T14+1456880186173970p15T15+3945615288257p18T16+2459621218p21T17+3369527p24T18+1336p27T19+p30T20 p^{9} T^{13} + 2681144587556984476 p^{12} T^{14} + 1456880186173970 p^{15} T^{15} + 3945615288257 p^{18} T^{16} + 2459621218 p^{21} T^{17} + 3369527 p^{24} T^{18} + 1336 p^{27} T^{19} + p^{30} T^{20}
89 1−1319T+3976549T2−3918029092T3+6926980930861T4−5155352728597703T5+7071873364160906092T6− 1 - 1319 T + 3976549 T^{2} - 3918029092 T^{3} + 6926980930861 T^{4} - 5155352728597703 T^{5} + 7071873364160906092 T^{6} - 39 ⁣⋯ ⁣8939\!\cdots\!89T7+ T^{7} + 50 ⁣⋯ ⁣0250\!\cdots\!02T8− T^{8} - 22 ⁣⋯ ⁣9322\!\cdots\!93T9+ T^{9} + 33 ⁣⋯ ⁣9033\!\cdots\!90T10− T^{10} - 22 ⁣⋯ ⁣9322\!\cdots\!93p3T11+ p^{3} T^{11} + 50 ⁣⋯ ⁣0250\!\cdots\!02p6T12− p^{6} T^{12} - 39 ⁣⋯ ⁣8939\!\cdots\!89p9T13+7071873364160906092p12T14−5155352728597703p15T15+6926980930861p18T16−3918029092p21T17+3976549p24T18−1319p27T19+p30T20 p^{9} T^{13} + 7071873364160906092 p^{12} T^{14} - 5155352728597703 p^{15} T^{15} + 6926980930861 p^{18} T^{16} - 3918029092 p^{21} T^{17} + 3976549 p^{24} T^{18} - 1319 p^{27} T^{19} + p^{30} T^{20}
97 1+2034T+7702255T2+12545817282T3+27183401753561T4+36890098418368256T5+58879246406538031644T6+ 1 + 2034 T + 7702255 T^{2} + 12545817282 T^{3} + 27183401753561 T^{4} + 36890098418368256 T^{5} + 58879246406538031644 T^{6} + 67 ⁣⋯ ⁣0467\!\cdots\!04T7+ T^{7} + 87 ⁣⋯ ⁣5487\!\cdots\!54T8+ T^{8} + 86 ⁣⋯ ⁣4086\!\cdots\!40T9+ T^{9} + 93 ⁣⋯ ⁣7493\!\cdots\!74T10+ T^{10} + 86 ⁣⋯ ⁣4086\!\cdots\!40p3T11+ p^{3} T^{11} + 87 ⁣⋯ ⁣5487\!\cdots\!54p6T12+ p^{6} T^{12} + 67 ⁣⋯ ⁣0467\!\cdots\!04p9T13+58879246406538031644p12T14+36890098418368256p15T15+27183401753561p18T16+12545817282p21T17+7702255p24T18+2034p27T19+p30T20 p^{9} T^{13} + 58879246406538031644 p^{12} T^{14} + 36890098418368256 p^{15} T^{15} + 27183401753561 p^{18} T^{16} + 12545817282 p^{21} T^{17} + 7702255 p^{24} T^{18} + 2034 p^{27} T^{19} + p^{30} T^{20}
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   L(s)=∏p ∏j=120(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−2.68735934564498486900759643678, −2.67614198966311015480956108344, −2.63672413388383592774629294101, −2.41173021018207672804971366778, −2.39489336974483060965063331190, −2.27220556798463851363445420647, −2.15668825733213665793625984049, −1.98248773648992568286429927967, −1.72800482282523772139055547022, −1.64180340462305514584450984033, −1.48705739979872738494563196773, −1.45932405887281340516049082699, −1.42107431494537558318352809265, −1.34125288101139308364089839747, −1.32996938391108176283983052108, −1.05754814945387658644118699200, −0.833559509572900423824944043485, −0.59360884035867361558856459438, −0.54745498461155909374413355612, −0.48258235850975317878369964175, −0.44804784250636344828325355484, −0.36577161397184355739453548989, −0.30022125825590516828855678210, −0.27467808573984025906871897129, −0.20083008966340051859894687835, 0.20083008966340051859894687835, 0.27467808573984025906871897129, 0.30022125825590516828855678210, 0.36577161397184355739453548989, 0.44804784250636344828325355484, 0.48258235850975317878369964175, 0.54745498461155909374413355612, 0.59360884035867361558856459438, 0.833559509572900423824944043485, 1.05754814945387658644118699200, 1.32996938391108176283983052108, 1.34125288101139308364089839747, 1.42107431494537558318352809265, 1.45932405887281340516049082699, 1.48705739979872738494563196773, 1.64180340462305514584450984033, 1.72800482282523772139055547022, 1.98248773648992568286429927967, 2.15668825733213665793625984049, 2.27220556798463851363445420647, 2.39489336974483060965063331190, 2.41173021018207672804971366778, 2.63672413388383592774629294101, 2.67614198966311015480956108344, 2.68735934564498486900759643678

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.