↑

Properties

Label 18-2523e9-1.1-c1e9-0-1
Degree 1818
Conductor 4.142×10304.142\times 10^{30}
Sign 11
Analytic cond. 5.46700×10115.46700\times 10^{11}
Root an. cond. 4.488454.48845
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 9·3-s − 3·4-s + 9·6-s + 5·7-s + 45·9-s + 3·11-s − 27·12-s + 5·13-s + 5·14-s + 16-s + 16·17-s + 45·18-s − 19-s + 45·21-s + 3·22-s − 10·23-s − 18·25-s + 5·26-s + 165·27-s − 15·28-s − 4·31-s − 11·32-s + 27·33-s + 16·34-s − 135·36-s + 25·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 5.19·3-s − 3/2·4-s + 3.67·6-s + 1.88·7-s + 15·9-s + 0.904·11-s − 7.79·12-s + 1.38·13-s + 1.33·14-s + 1/4·16-s + 3.88·17-s + 10.6·18-s − 0.229·19-s + 9.81·21-s + 0.639·22-s − 2.08·23-s − 3.59·25-s + 0.980·26-s + 31.7·27-s − 2.83·28-s − 0.718·31-s − 1.94·32-s + 4.70·33-s + 2.74·34-s − 22.5·36-s + 4.10·37-s + ⋯

Functional equation

Λ(s)=((39⋅2918)s/2 ΓC(s)9 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((39⋅2918)s/2 ΓC(s+1/2)9 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1818
Conductor: 39⋅29183^{9} \cdot 29^{18}
Sign: 11
Analytic conductor: 5.46700×10115.46700\times 10^{11}
Root analytic conductor: 4.488454.48845
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (18, 39⋅2918, ( :[1/2]9), 1)(18,\ 3^{9} \cdot 29^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 784.4245991784.4245991
L(12)L(\frac12) ≈\approx 784.4245991784.4245991
L(32)L(\frac{3}{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−T)9 ( 1 - T )^{9}
29 1 1
good2 1−T+p2T2−7T3+9pT4−27T5+55T6−39pT7+67pT8−183T9+67p2T10−39p3T11+55p3T12−27p4T13+9p6T14−7p6T15+p9T16−p8T17+p9T18 1 - T + p^{2} T^{2} - 7 T^{3} + 9 p T^{4} - 27 T^{5} + 55 T^{6} - 39 p T^{7} + 67 p T^{8} - 183 T^{9} + 67 p^{2} T^{10} - 39 p^{3} T^{11} + 55 p^{3} T^{12} - 27 p^{4} T^{13} + 9 p^{6} T^{14} - 7 p^{6} T^{15} + p^{9} T^{16} - p^{8} T^{17} + p^{9} T^{18}
5 1+18T2+24T3+177T4+337T5+1492T6+2633T7+9457T8+15649T9+9457pT10+2633p2T11+1492p3T12+337p4T13+177p5T14+24p6T15+18p7T16+p9T18 1 + 18 T^{2} + 24 T^{3} + 177 T^{4} + 337 T^{5} + 1492 T^{6} + 2633 T^{7} + 9457 T^{8} + 15649 T^{9} + 9457 p T^{10} + 2633 p^{2} T^{11} + 1492 p^{3} T^{12} + 337 p^{4} T^{13} + 177 p^{5} T^{14} + 24 p^{6} T^{15} + 18 p^{7} T^{16} + p^{9} T^{18}
7 1−5T+31T2−128T3+463T4−1405T5+4100T6−9956T7+25827T8−65596T9+25827pT10−9956p2T11+4100p3T12−1405p4T13+463p5T14−128p6T15+31p7T16−5p8T17+p9T18 1 - 5 T + 31 T^{2} - 128 T^{3} + 463 T^{4} - 1405 T^{5} + 4100 T^{6} - 9956 T^{7} + 25827 T^{8} - 65596 T^{9} + 25827 p T^{10} - 9956 p^{2} T^{11} + 4100 p^{3} T^{12} - 1405 p^{4} T^{13} + 463 p^{5} T^{14} - 128 p^{6} T^{15} + 31 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18}
11 1−3T+49T2−52T3+991T4+513T5+13154T6+27866T7+1147p2T8+445824T9+1147p3T10+27866p2T11+13154p3T12+513p4T13+991p5T14−52p6T15+49p7T16−3p8T17+p9T18 1 - 3 T + 49 T^{2} - 52 T^{3} + 991 T^{4} + 513 T^{5} + 13154 T^{6} + 27866 T^{7} + 1147 p^{2} T^{8} + 445824 T^{9} + 1147 p^{3} T^{10} + 27866 p^{2} T^{11} + 13154 p^{3} T^{12} + 513 p^{4} T^{13} + 991 p^{5} T^{14} - 52 p^{6} T^{15} + 49 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18}
13 1−5T+102T2−33pT3+4788T4−17120T5+135677T6−412188T7+2557299T8−6530585T9+2557299pT10−412188p2T11+135677p3T12−17120p4T13+4788p5T14−33p7T15+102p7T16−5p8T17+p9T18 1 - 5 T + 102 T^{2} - 33 p T^{3} + 4788 T^{4} - 17120 T^{5} + 135677 T^{6} - 412188 T^{7} + 2557299 T^{8} - 6530585 T^{9} + 2557299 p T^{10} - 412188 p^{2} T^{11} + 135677 p^{3} T^{12} - 17120 p^{4} T^{13} + 4788 p^{5} T^{14} - 33 p^{7} T^{15} + 102 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18}
17 1−16T+190T2−1544T3+11104T4−66663T5+376150T6−1868520T7+8820919T8−37225411T9+8820919pT10−1868520p2T11+376150p3T12−66663p4T13+11104p5T14−1544p6T15+190p7T16−16p8T17+p9T18 1 - 16 T + 190 T^{2} - 1544 T^{3} + 11104 T^{4} - 66663 T^{5} + 376150 T^{6} - 1868520 T^{7} + 8820919 T^{8} - 37225411 T^{9} + 8820919 p T^{10} - 1868520 p^{2} T^{11} + 376150 p^{3} T^{12} - 66663 p^{4} T^{13} + 11104 p^{5} T^{14} - 1544 p^{6} T^{15} + 190 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18}
19 1+T+144T2+154T3+9521T4+10329T5+382344T6+394400T7+10349226T8+9381584T9+10349226pT10+394400p2T11+382344p3T12+10329p4T13+9521p5T14+154p6T15+144p7T16+p8T17+p9T18 1 + T + 144 T^{2} + 154 T^{3} + 9521 T^{4} + 10329 T^{5} + 382344 T^{6} + 394400 T^{7} + 10349226 T^{8} + 9381584 T^{9} + 10349226 p T^{10} + 394400 p^{2} T^{11} + 382344 p^{3} T^{12} + 10329 p^{4} T^{13} + 9521 p^{5} T^{14} + 154 p^{6} T^{15} + 144 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18}
23 1+10T+8pT2+1416T3+15023T4+94178T5+734858T6+3853590T7+24114842T8+106498388T9+24114842pT10+3853590p2T11+734858p3T12+94178p4T13+15023p5T14+1416p6T15+8p8T16+10p8T17+p9T18 1 + 10 T + 8 p T^{2} + 1416 T^{3} + 15023 T^{4} + 94178 T^{5} + 734858 T^{6} + 3853590 T^{7} + 24114842 T^{8} + 106498388 T^{9} + 24114842 p T^{10} + 3853590 p^{2} T^{11} + 734858 p^{3} T^{12} + 94178 p^{4} T^{13} + 15023 p^{5} T^{14} + 1416 p^{6} T^{15} + 8 p^{8} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18}
31 1+4T+95T2+604T3+4724T4+31293T5+171357T6+961972T7+4798255T8+27630886T9+4798255pT10+961972p2T11+171357p3T12+31293p4T13+4724p5T14+604p6T15+95p7T16+4p8T17+p9T18 1 + 4 T + 95 T^{2} + 604 T^{3} + 4724 T^{4} + 31293 T^{5} + 171357 T^{6} + 961972 T^{7} + 4798255 T^{8} + 27630886 T^{9} + 4798255 p T^{10} + 961972 p^{2} T^{11} + 171357 p^{3} T^{12} + 31293 p^{4} T^{13} + 4724 p^{5} T^{14} + 604 p^{6} T^{15} + 95 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18}
37 1−25T+476T2−6436T3+74730T4−726039T5+6338716T6−48776974T7+344118462T8−2179787403T9+344118462pT10−48776974p2T11+6338716p3T12−726039p4T13+74730p5T14−6436p6T15+476p7T16−25p8T17+p9T18 1 - 25 T + 476 T^{2} - 6436 T^{3} + 74730 T^{4} - 726039 T^{5} + 6338716 T^{6} - 48776974 T^{7} + 344118462 T^{8} - 2179787403 T^{9} + 344118462 p T^{10} - 48776974 p^{2} T^{11} + 6338716 p^{3} T^{12} - 726039 p^{4} T^{13} + 74730 p^{5} T^{14} - 6436 p^{6} T^{15} + 476 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18}
41 1−34T+788T2−12970T3+175640T4−1971845T5+19301782T6−164588816T7+1253221727T8−8463182695T9+1253221727pT10−164588816p2T11+19301782p3T12−1971845p4T13+175640p5T14−12970p6T15+788p7T16−34p8T17+p9T18 1 - 34 T + 788 T^{2} - 12970 T^{3} + 175640 T^{4} - 1971845 T^{5} + 19301782 T^{6} - 164588816 T^{7} + 1253221727 T^{8} - 8463182695 T^{9} + 1253221727 p T^{10} - 164588816 p^{2} T^{11} + 19301782 p^{3} T^{12} - 1971845 p^{4} T^{13} + 175640 p^{5} T^{14} - 12970 p^{6} T^{15} + 788 p^{7} T^{16} - 34 p^{8} T^{17} + p^{9} T^{18}
43 1−12T+217T2−2702T3+28804T4−281927T5+2520869T6−19935336T7+147994305T8−1019469718T9+147994305pT10−19935336p2T11+2520869p3T12−281927p4T13+28804p5T14−2702p6T15+217p7T16−12p8T17+p9T18 1 - 12 T + 217 T^{2} - 2702 T^{3} + 28804 T^{4} - 281927 T^{5} + 2520869 T^{6} - 19935336 T^{7} + 147994305 T^{8} - 1019469718 T^{9} + 147994305 p T^{10} - 19935336 p^{2} T^{11} + 2520869 p^{3} T^{12} - 281927 p^{4} T^{13} + 28804 p^{5} T^{14} - 2702 p^{6} T^{15} + 217 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18}
47 1−8T+268T2−2089T3+38781T4−267015T5+3604314T6−21845867T7+235040160T8−1222532802T9+235040160pT10−21845867p2T11+3604314p3T12−267015p4T13+38781p5T14−2089p6T15+268p7T16−8p8T17+p9T18 1 - 8 T + 268 T^{2} - 2089 T^{3} + 38781 T^{4} - 267015 T^{5} + 3604314 T^{6} - 21845867 T^{7} + 235040160 T^{8} - 1222532802 T^{9} + 235040160 p T^{10} - 21845867 p^{2} T^{11} + 3604314 p^{3} T^{12} - 267015 p^{4} T^{13} + 38781 p^{5} T^{14} - 2089 p^{6} T^{15} + 268 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18}
53 1+32T+777T2+13556T3+199931T4+2475132T5+27002767T6+258706391T7+2223873503T8+17047663403T9+2223873503pT10+258706391p2T11+27002767p3T12+2475132p4T13+199931p5T14+13556p6T15+777p7T16+32p8T17+p9T18 1 + 32 T + 777 T^{2} + 13556 T^{3} + 199931 T^{4} + 2475132 T^{5} + 27002767 T^{6} + 258706391 T^{7} + 2223873503 T^{8} + 17047663403 T^{9} + 2223873503 p T^{10} + 258706391 p^{2} T^{11} + 27002767 p^{3} T^{12} + 2475132 p^{4} T^{13} + 199931 p^{5} T^{14} + 13556 p^{6} T^{15} + 777 p^{7} T^{16} + 32 p^{8} T^{17} + p^{9} T^{18}
59 1−10T+263T2−28pT3+30369T4−110304T5+2000488T6−1128580T7+96017121T8+142377228T9+96017121pT10−1128580p2T11+2000488p3T12−110304p4T13+30369p5T14−28p7T15+263p7T16−10p8T17+p9T18 1 - 10 T + 263 T^{2} - 28 p T^{3} + 30369 T^{4} - 110304 T^{5} + 2000488 T^{6} - 1128580 T^{7} + 96017121 T^{8} + 142377228 T^{9} + 96017121 p T^{10} - 1128580 p^{2} T^{11} + 2000488 p^{3} T^{12} - 110304 p^{4} T^{13} + 30369 p^{5} T^{14} - 28 p^{7} T^{15} + 263 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18}
61 1−51T+1613T2−36289T3+10652pT4−9576025T5+1971498pT6−1302879380T7+12352053601T8−102748700237T9+12352053601pT10−1302879380p2T11+1971498p4T12−9576025p4T13+10652p6T14−36289p6T15+1613p7T16−51p8T17+p9T18 1 - 51 T + 1613 T^{2} - 36289 T^{3} + 10652 p T^{4} - 9576025 T^{5} + 1971498 p T^{6} - 1302879380 T^{7} + 12352053601 T^{8} - 102748700237 T^{9} + 12352053601 p T^{10} - 1302879380 p^{2} T^{11} + 1971498 p^{4} T^{12} - 9576025 p^{4} T^{13} + 10652 p^{6} T^{14} - 36289 p^{6} T^{15} + 1613 p^{7} T^{16} - 51 p^{8} T^{17} + p^{9} T^{18}
67 1−7T+271T2−2196T3+43720T4−328746T5+4849035T6−33932684T7+408787801T8−2583256998T9+408787801pT10−33932684p2T11+4849035p3T12−328746p4T13+43720p5T14−2196p6T15+271p7T16−7p8T17+p9T18 1 - 7 T + 271 T^{2} - 2196 T^{3} + 43720 T^{4} - 328746 T^{5} + 4849035 T^{6} - 33932684 T^{7} + 408787801 T^{8} - 2583256998 T^{9} + 408787801 p T^{10} - 33932684 p^{2} T^{11} + 4849035 p^{3} T^{12} - 328746 p^{4} T^{13} + 43720 p^{5} T^{14} - 2196 p^{6} T^{15} + 271 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18}
71 1−7T+403T2−2578T3+79996T4−463984T5+10354421T6−54242872T7+972589151T8−4513565462T9+972589151pT10−54242872p2T11+10354421p3T12−463984p4T13+79996p5T14−2578p6T15+403p7T16−7p8T17+p9T18 1 - 7 T + 403 T^{2} - 2578 T^{3} + 79996 T^{4} - 463984 T^{5} + 10354421 T^{6} - 54242872 T^{7} + 972589151 T^{8} - 4513565462 T^{9} + 972589151 p T^{10} - 54242872 p^{2} T^{11} + 10354421 p^{3} T^{12} - 463984 p^{4} T^{13} + 79996 p^{5} T^{14} - 2578 p^{6} T^{15} + 403 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18}
73 1−17T+401T2−6426T3+92991T4−1179384T5+13953041T6−144174105T7+1416502918T8−12512366475T9+1416502918pT10−144174105p2T11+13953041p3T12−1179384p4T13+92991p5T14−6426p6T15+401p7T16−17p8T17+p9T18 1 - 17 T + 401 T^{2} - 6426 T^{3} + 92991 T^{4} - 1179384 T^{5} + 13953041 T^{6} - 144174105 T^{7} + 1416502918 T^{8} - 12512366475 T^{9} + 1416502918 p T^{10} - 144174105 p^{2} T^{11} + 13953041 p^{3} T^{12} - 1179384 p^{4} T^{13} + 92991 p^{5} T^{14} - 6426 p^{6} T^{15} + 401 p^{7} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18}
79 1−13T+543T2−6692T3+144736T4−1578690T5+24184789T6−226132206T7+2738544831T8−21609843142T9+2738544831pT10−226132206p2T11+24184789p3T12−1578690p4T13+144736p5T14−6692p6T15+543p7T16−13p8T17+p9T18 1 - 13 T + 543 T^{2} - 6692 T^{3} + 144736 T^{4} - 1578690 T^{5} + 24184789 T^{6} - 226132206 T^{7} + 2738544831 T^{8} - 21609843142 T^{9} + 2738544831 p T^{10} - 226132206 p^{2} T^{11} + 24184789 p^{3} T^{12} - 1578690 p^{4} T^{13} + 144736 p^{5} T^{14} - 6692 p^{6} T^{15} + 543 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18}
83 1+31T+1023T2+20178T3+391292T4+5722828T5+81071705T6+933533148T7+10375704955T8+96245293542T9+10375704955pT10+933533148p2T11+81071705p3T12+5722828p4T13+391292p5T14+20178p6T15+1023p7T16+31p8T17+p9T18 1 + 31 T + 1023 T^{2} + 20178 T^{3} + 391292 T^{4} + 5722828 T^{5} + 81071705 T^{6} + 933533148 T^{7} + 10375704955 T^{8} + 96245293542 T^{9} + 10375704955 p T^{10} + 933533148 p^{2} T^{11} + 81071705 p^{3} T^{12} + 5722828 p^{4} T^{13} + 391292 p^{5} T^{14} + 20178 p^{6} T^{15} + 1023 p^{7} T^{16} + 31 p^{8} T^{17} + p^{9} T^{18}
89 1+32T+894T2+16539T3+282952T4+3946842T5+51974585T6+593765484T7+6445461922T8+62230806243T9+6445461922pT10+593765484p2T11+51974585p3T12+3946842p4T13+282952p5T14+16539p6T15+894p7T16+32p8T17+p9T18 1 + 32 T + 894 T^{2} + 16539 T^{3} + 282952 T^{4} + 3946842 T^{5} + 51974585 T^{6} + 593765484 T^{7} + 6445461922 T^{8} + 62230806243 T^{9} + 6445461922 p T^{10} + 593765484 p^{2} T^{11} + 51974585 p^{3} T^{12} + 3946842 p^{4} T^{13} + 282952 p^{5} T^{14} + 16539 p^{6} T^{15} + 894 p^{7} T^{16} + 32 p^{8} T^{17} + p^{9} T^{18}
97 1−16T+446T2−5487T3+94470T4−931906T5+13217605T6−110931128T7+1428628652T8−11184892691T9+1428628652pT10−110931128p2T11+13217605p3T12−931906p4T13+94470p5T14−5487p6T15+446p7T16−16p8T17+p9T18 1 - 16 T + 446 T^{2} - 5487 T^{3} + 94470 T^{4} - 931906 T^{5} + 13217605 T^{6} - 110931128 T^{7} + 1428628652 T^{8} - 11184892691 T^{9} + 1428628652 p T^{10} - 110931128 p^{2} T^{11} + 13217605 p^{3} T^{12} - 931906 p^{4} T^{13} + 94470 p^{5} T^{14} - 5487 p^{6} T^{15} + 446 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18}
show more
show less
   L(s)=∏p ∏j=118(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.53269985047208945955943788342, −3.35728049878546155691485862163, −3.31301686238696893621401657194, −3.14290691604254052858212129524, −2.83153064707314165880836621751, −2.80586452443707872763580330191, −2.74718352081279154359818833385, −2.61124287712323453283774993431, −2.54721740341077306209339293743, −2.48018483363078393814937966698, −2.11288301615040538975344383317, −2.08690429227919172411047336554, −2.07395145273429544543306432019, −2.00507173398224234399566099927, −2.00387167805507805308711083837, −1.75658108574976970475325510291, −1.73961659817377672090806975657, −1.36823080135923419487002372820, −1.22037220261402304154742936630, −1.20156259228613038576132505057, −1.06555775352255085648153426782, −0.848958585214715597394713242811, −0.806989254111012787227852243433, −0.65618362649692908041785953673, −0.41215820423216672000067788353, 0.41215820423216672000067788353, 0.65618362649692908041785953673, 0.806989254111012787227852243433, 0.848958585214715597394713242811, 1.06555775352255085648153426782, 1.20156259228613038576132505057, 1.22037220261402304154742936630, 1.36823080135923419487002372820, 1.73961659817377672090806975657, 1.75658108574976970475325510291, 2.00387167805507805308711083837, 2.00507173398224234399566099927, 2.07395145273429544543306432019, 2.08690429227919172411047336554, 2.11288301615040538975344383317, 2.48018483363078393814937966698, 2.54721740341077306209339293743, 2.61124287712323453283774993431, 2.74718352081279154359818833385, 2.80586452443707872763580330191, 2.83153064707314165880836621751, 3.14290691604254052858212129524, 3.31301686238696893621401657194, 3.35728049878546155691485862163, 3.53269985047208945955943788342

Graph of the ZZ-function along the critical line

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