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Properties

Label 16-31e16-1.1-c1e8-0-14
Degree 1616
Conductor 7.274×10237.274\times 10^{23}
Sign 11
Analytic cond. 1.20227×1071.20227\times 10^{7}
Root an. cond. 2.770132.77013
Motivic weight 11
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  + 2·2-s + 3·3-s − 2·4-s + 3·5-s + 6·6-s − 2·7-s − 11·8-s − 5·9-s + 6·10-s + 18·11-s − 6·12-s + 8·13-s − 4·14-s + 9·15-s − 11·16-s + 14·17-s − 10·18-s − 6·19-s − 6·20-s − 6·21-s + 36·22-s + 22·23-s − 33·24-s − 9·25-s + 16·26-s − 21·27-s + 4·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s − 4-s + 1.34·5-s + 2.44·6-s − 0.755·7-s − 3.88·8-s − 5/3·9-s + 1.89·10-s + 5.42·11-s − 1.73·12-s + 2.21·13-s − 1.06·14-s + 2.32·15-s − 2.75·16-s + 3.39·17-s − 2.35·18-s − 1.37·19-s − 1.34·20-s − 1.30·21-s + 7.67·22-s + 4.58·23-s − 6.73·24-s − 9/5·25-s + 3.13·26-s − 4.04·27-s + 0.755·28-s + ⋯

Functional equation

Λ(s)=((3116)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((3116)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 311631^{16}
Sign: 11
Analytic conductor: 1.20227×1071.20227\times 10^{7}
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 3116, ( :[1/2]8), 1)(16,\ 31^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 42.8015588142.80155881
L(12)L(\frac12) ≈\approx 42.8015588142.80155881
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)
bad31 1 1
good2 1−pT+3pT2−5T3+11T4−T5+15T6+9T7+27T8+9pT9+15p2T10−p3T11+11p4T12−5p5T13+3p7T14−p8T15+p8T16 1 - p T + 3 p T^{2} - 5 T^{3} + 11 T^{4} - T^{5} + 15 T^{6} + 9 T^{7} + 27 T^{8} + 9 p T^{9} + 15 p^{2} T^{10} - p^{3} T^{11} + 11 p^{4} T^{12} - 5 p^{5} T^{13} + 3 p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16}
3 1−pT+14T2−4p2T3+37pT4−26p2T5+560T6−335pT7+1987T8−335p2T9+560p2T10−26p5T11+37p5T12−4p7T13+14p6T14−p8T15+p8T16 1 - p T + 14 T^{2} - 4 p^{2} T^{3} + 37 p T^{4} - 26 p^{2} T^{5} + 560 T^{6} - 335 p T^{7} + 1987 T^{8} - 335 p^{2} T^{9} + 560 p^{2} T^{10} - 26 p^{5} T^{11} + 37 p^{5} T^{12} - 4 p^{7} T^{13} + 14 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16}
5 1−3T+18T2−36T3+31pT4−261T5+993T6−1428T7+5151T8−1428pT9+993p2T10−261p3T11+31p5T12−36p5T13+18p6T14−3p7T15+p8T16 1 - 3 T + 18 T^{2} - 36 T^{3} + 31 p T^{4} - 261 T^{5} + 993 T^{6} - 1428 T^{7} + 5151 T^{8} - 1428 p T^{9} + 993 p^{2} T^{10} - 261 p^{3} T^{11} + 31 p^{5} T^{12} - 36 p^{5} T^{13} + 18 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16}
7 1+2T+31T2+60T3+506T4+881T5+5550T6+8441T7+44837T8+8441pT9+5550p2T10+881p3T11+506p4T12+60p5T13+31p6T14+2p7T15+p8T16 1 + 2 T + 31 T^{2} + 60 T^{3} + 506 T^{4} + 881 T^{5} + 5550 T^{6} + 8441 T^{7} + 44837 T^{8} + 8441 p T^{9} + 5550 p^{2} T^{10} + 881 p^{3} T^{11} + 506 p^{4} T^{12} + 60 p^{5} T^{13} + 31 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
11 1−18T+219T2−1875T3+12998T4−73485T5+354666T6−132522pT7+5210553T8−132522p2T9+354666p2T10−73485p3T11+12998p4T12−1875p5T13+219p6T14−18p7T15+p8T16 1 - 18 T + 219 T^{2} - 1875 T^{3} + 12998 T^{4} - 73485 T^{5} + 354666 T^{6} - 132522 p T^{7} + 5210553 T^{8} - 132522 p^{2} T^{9} + 354666 p^{2} T^{10} - 73485 p^{3} T^{11} + 12998 p^{4} T^{12} - 1875 p^{5} T^{13} + 219 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16}
13 1−8T+100T2−567T3+4139T4−18554T5+100344T6−368819T7+1591961T8−368819pT9+100344p2T10−18554p3T11+4139p4T12−567p5T13+100p6T14−8p7T15+p8T16 1 - 8 T + 100 T^{2} - 567 T^{3} + 4139 T^{4} - 18554 T^{5} + 100344 T^{6} - 368819 T^{7} + 1591961 T^{8} - 368819 p T^{9} + 100344 p^{2} T^{10} - 18554 p^{3} T^{11} + 4139 p^{4} T^{12} - 567 p^{5} T^{13} + 100 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
17 1−14T+171T2−1412T3+10571T4−64147T5+358770T6−1712910T7+7592697T8−1712910pT9+358770p2T10−64147p3T11+10571p4T12−1412p5T13+171p6T14−14p7T15+p8T16 1 - 14 T + 171 T^{2} - 1412 T^{3} + 10571 T^{4} - 64147 T^{5} + 358770 T^{6} - 1712910 T^{7} + 7592697 T^{8} - 1712910 p T^{9} + 358770 p^{2} T^{10} - 64147 p^{3} T^{11} + 10571 p^{4} T^{12} - 1412 p^{5} T^{13} + 171 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16}
19 1+6T+126T2+550T3+6723T4+22465T5+213499T6+577881T7+4727383T8+577881pT9+213499p2T10+22465p3T11+6723p4T12+550p5T13+126p6T14+6p7T15+p8T16 1 + 6 T + 126 T^{2} + 550 T^{3} + 6723 T^{4} + 22465 T^{5} + 213499 T^{6} + 577881 T^{7} + 4727383 T^{8} + 577881 p T^{9} + 213499 p^{2} T^{10} + 22465 p^{3} T^{11} + 6723 p^{4} T^{12} + 550 p^{5} T^{13} + 126 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
23 1−22T+372T2−4321T3+42281T4−334832T5+2312310T6−13546893T7+70125249T8−13546893pT9+2312310p2T10−334832p3T11+42281p4T12−4321p5T13+372p6T14−22p7T15+p8T16 1 - 22 T + 372 T^{2} - 4321 T^{3} + 42281 T^{4} - 334832 T^{5} + 2312310 T^{6} - 13546893 T^{7} + 70125249 T^{8} - 13546893 p T^{9} + 2312310 p^{2} T^{10} - 334832 p^{3} T^{11} + 42281 p^{4} T^{12} - 4321 p^{5} T^{13} + 372 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16}
29 1−12T+228T2−2067T3+22391T4−160170T5+1262877T6−7289670T7+45291747T8−7289670pT9+1262877p2T10−160170p3T11+22391p4T12−2067p5T13+228p6T14−12p7T15+p8T16 1 - 12 T + 228 T^{2} - 2067 T^{3} + 22391 T^{4} - 160170 T^{5} + 1262877 T^{6} - 7289670 T^{7} + 45291747 T^{8} - 7289670 p T^{9} + 1262877 p^{2} T^{10} - 160170 p^{3} T^{11} + 22391 p^{4} T^{12} - 2067 p^{5} T^{13} + 228 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16}
37 1+8T+184T2+1003T3+14093T4+50978T5+642206T6+1564129T7+23810023T8+1564129pT9+642206p2T10+50978p3T11+14093p4T12+1003p5T13+184p6T14+8p7T15+p8T16 1 + 8 T + 184 T^{2} + 1003 T^{3} + 14093 T^{4} + 50978 T^{5} + 642206 T^{6} + 1564129 T^{7} + 23810023 T^{8} + 1564129 p T^{9} + 642206 p^{2} T^{10} + 50978 p^{3} T^{11} + 14093 p^{4} T^{12} + 1003 p^{5} T^{13} + 184 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16}
41 1+22T+498T2+6811T3+89027T4+877490T5+8169387T6+61298622T7+433574745T8+61298622pT9+8169387p2T10+877490p3T11+89027p4T12+6811p5T13+498p6T14+22p7T15+p8T16 1 + 22 T + 498 T^{2} + 6811 T^{3} + 89027 T^{4} + 877490 T^{5} + 8169387 T^{6} + 61298622 T^{7} + 433574745 T^{8} + 61298622 p T^{9} + 8169387 p^{2} T^{10} + 877490 p^{3} T^{11} + 89027 p^{4} T^{12} + 6811 p^{5} T^{13} + 498 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16}
43 1+2T+205T2+403T3+21134T4+40961T5+1453424T6+2649346T7+72598441T8+2649346pT9+1453424p2T10+40961p3T11+21134p4T12+403p5T13+205p6T14+2p7T15+p8T16 1 + 2 T + 205 T^{2} + 403 T^{3} + 21134 T^{4} + 40961 T^{5} + 1453424 T^{6} + 2649346 T^{7} + 72598441 T^{8} + 2649346 p T^{9} + 1453424 p^{2} T^{10} + 40961 p^{3} T^{11} + 21134 p^{4} T^{12} + 403 p^{5} T^{13} + 205 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
47 1+18T+411T2+4965T3+66911T4+618324T5+6152625T6+45556584T7+358374597T8+45556584pT9+6152625p2T10+618324p3T11+66911p4T12+4965p5T13+411p6T14+18p7T15+p8T16 1 + 18 T + 411 T^{2} + 4965 T^{3} + 66911 T^{4} + 618324 T^{5} + 6152625 T^{6} + 45556584 T^{7} + 358374597 T^{8} + 45556584 p T^{9} + 6152625 p^{2} T^{10} + 618324 p^{3} T^{11} + 66911 p^{4} T^{12} + 4965 p^{5} T^{13} + 411 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16}
53 1−6T+244T2−1605T3+30376T4−203598T5+2505355T6−16037007T7+152391427T8−16037007pT9+2505355p2T10−203598p3T11+30376p4T12−1605p5T13+244p6T14−6p7T15+p8T16 1 - 6 T + 244 T^{2} - 1605 T^{3} + 30376 T^{4} - 203598 T^{5} + 2505355 T^{6} - 16037007 T^{7} + 152391427 T^{8} - 16037007 p T^{9} + 2505355 p^{2} T^{10} - 203598 p^{3} T^{11} + 30376 p^{4} T^{12} - 1605 p^{5} T^{13} + 244 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
59 1+4T+309T2+1189T3+45401T4+174557T5+4316301T6+15922002T7+295944207T8+15922002pT9+4316301p2T10+174557p3T11+45401p4T12+1189p5T13+309p6T14+4p7T15+p8T16 1 + 4 T + 309 T^{2} + 1189 T^{3} + 45401 T^{4} + 174557 T^{5} + 4316301 T^{6} + 15922002 T^{7} + 295944207 T^{8} + 15922002 p T^{9} + 4316301 p^{2} T^{10} + 174557 p^{3} T^{11} + 45401 p^{4} T^{12} + 1189 p^{5} T^{13} + 309 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
61 1−30T+776T2−13665T3+208005T4−2593710T5+28390399T6−268109040T7+2240276459T8−268109040pT9+28390399p2T10−2593710p3T11+208005p4T12−13665p5T13+776p6T14−30p7T15+p8T16 1 - 30 T + 776 T^{2} - 13665 T^{3} + 208005 T^{4} - 2593710 T^{5} + 28390399 T^{6} - 268109040 T^{7} + 2240276459 T^{8} - 268109040 p T^{9} + 28390399 p^{2} T^{10} - 2593710 p^{3} T^{11} + 208005 p^{4} T^{12} - 13665 p^{5} T^{13} + 776 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16}
67 1+13T+427T2+4781T3+85148T4+816694T5+10335824T6+84073193T7+837177553T8+84073193pT9+10335824p2T10+816694p3T11+85148p4T12+4781p5T13+427p6T14+13p7T15+p8T16 1 + 13 T + 427 T^{2} + 4781 T^{3} + 85148 T^{4} + 816694 T^{5} + 10335824 T^{6} + 84073193 T^{7} + 837177553 T^{8} + 84073193 p T^{9} + 10335824 p^{2} T^{10} + 816694 p^{3} T^{11} + 85148 p^{4} T^{12} + 4781 p^{5} T^{13} + 427 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16}
71 1+T+465T2+325T3+99635T4+48353T5+12913848T6+4606650T7+1111294875T8+4606650pT9+12913848p2T10+48353p3T11+99635p4T12+325p5T13+465p6T14+p7T15+p8T16 1 + T + 465 T^{2} + 325 T^{3} + 99635 T^{4} + 48353 T^{5} + 12913848 T^{6} + 4606650 T^{7} + 1111294875 T^{8} + 4606650 p T^{9} + 12913848 p^{2} T^{10} + 48353 p^{3} T^{11} + 99635 p^{4} T^{12} + 325 p^{5} T^{13} + 465 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16}
73 1−2T+280T2−423T3+39254T4−42716T5+3669099T6−2910611T7+282548711T8−2910611pT9+3669099p2T10−42716p3T11+39254p4T12−423p5T13+280p6T14−2p7T15+p8T16 1 - 2 T + 280 T^{2} - 423 T^{3} + 39254 T^{4} - 42716 T^{5} + 3669099 T^{6} - 2910611 T^{7} + 282548711 T^{8} - 2910611 p T^{9} + 3669099 p^{2} T^{10} - 42716 p^{3} T^{11} + 39254 p^{4} T^{12} - 423 p^{5} T^{13} + 280 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
79 1−8T+343T2−2748T3+65666T4−478310T5+8349177T6−54496370T7+769049897T8−54496370pT9+8349177p2T10−478310p3T11+65666p4T12−2748p5T13+343p6T14−8p7T15+p8T16 1 - 8 T + 343 T^{2} - 2748 T^{3} + 65666 T^{4} - 478310 T^{5} + 8349177 T^{6} - 54496370 T^{7} + 769049897 T^{8} - 54496370 p T^{9} + 8349177 p^{2} T^{10} - 478310 p^{3} T^{11} + 65666 p^{4} T^{12} - 2748 p^{5} T^{13} + 343 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
83 1−39T+12pT2−18231T3+273488T4−3449334T5+38635404T6−390596853T7+3688239693T8−390596853pT9+38635404p2T10−3449334p3T11+273488p4T12−18231p5T13+12p7T14−39p7T15+p8T16 1 - 39 T + 12 p T^{2} - 18231 T^{3} + 273488 T^{4} - 3449334 T^{5} + 38635404 T^{6} - 390596853 T^{7} + 3688239693 T^{8} - 390596853 p T^{9} + 38635404 p^{2} T^{10} - 3449334 p^{3} T^{11} + 273488 p^{4} T^{12} - 18231 p^{5} T^{13} + 12 p^{7} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16}
89 1−27T+807T2−15033T3+266447T4−3727035T5+48700278T6−532857324T7+5459588145T8−532857324pT9+48700278p2T10−3727035p3T11+266447p4T12−15033p5T13+807p6T14−27p7T15+p8T16 1 - 27 T + 807 T^{2} - 15033 T^{3} + 266447 T^{4} - 3727035 T^{5} + 48700278 T^{6} - 532857324 T^{7} + 5459588145 T^{8} - 532857324 p T^{9} + 48700278 p^{2} T^{10} - 3727035 p^{3} T^{11} + 266447 p^{4} T^{12} - 15033 p^{5} T^{13} + 807 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16}
97 1−34T+1021T2−20487T3+372881T4−5460517T5+73647420T6−843517135T7+8954673947T8−843517135pT9+73647420p2T10−5460517p3T11+372881p4T12−20487p5T13+1021p6T14−34p7T15+p8T16 1 - 34 T + 1021 T^{2} - 20487 T^{3} + 372881 T^{4} - 5460517 T^{5} + 73647420 T^{6} - 843517135 T^{7} + 8954673947 T^{8} - 843517135 p T^{9} + 73647420 p^{2} T^{10} - 5460517 p^{3} T^{11} + 372881 p^{4} T^{12} - 20487 p^{5} T^{13} + 1021 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.48922036255695188032153995746, −3.90440327798996540820770290020, −3.85644654283247669571367407202, −3.67785749657087078758425042079, −3.61225763734995469468947103785, −3.55246081584157521437001666567, −3.52161221072777681190294212969, −3.51512656782749023188541822393, −3.21565094478984252232131943120, −3.18216519625795522332704430782, −3.12208589020671323357587230479, −3.10796975895595825101672747725, −2.99888128042516508548917096363, −2.73718964775739896649560550894, −2.28106842267315120489838433733, −2.07670455890525745557489366450, −1.91189006466009234453826698555, −1.90785407613630421997222776427, −1.89096130756384857595321329742, −1.53075198023429447349881932914, −1.24713037714290168964586397242, −1.06112232391071405361692778038, −0.74250389482280370000216607371, −0.64716136886203316775996400597, −0.59097615621094013380322393746, 0.59097615621094013380322393746, 0.64716136886203316775996400597, 0.74250389482280370000216607371, 1.06112232391071405361692778038, 1.24713037714290168964586397242, 1.53075198023429447349881932914, 1.89096130756384857595321329742, 1.90785407613630421997222776427, 1.91189006466009234453826698555, 2.07670455890525745557489366450, 2.28106842267315120489838433733, 2.73718964775739896649560550894, 2.99888128042516508548917096363, 3.10796975895595825101672747725, 3.12208589020671323357587230479, 3.18216519625795522332704430782, 3.21565094478984252232131943120, 3.51512656782749023188541822393, 3.52161221072777681190294212969, 3.55246081584157521437001666567, 3.61225763734995469468947103785, 3.67785749657087078758425042079, 3.85644654283247669571367407202, 3.90440327798996540820770290020, 4.48922036255695188032153995746

Graph of the ZZ-function along the critical line

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