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Properties

Label 16-31e16-1.1-c1e8-0-9
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3-s + 8·4-s + 6·5-s − 4·6-s + 3·7-s + 18·8-s − 2·9-s + 24·10-s − 8·11-s − 8·12-s + 9·13-s + 12·14-s − 6·15-s + 34·16-s − 7·17-s − 8·18-s + 5·19-s + 48·20-s − 3·21-s − 32·22-s − 18·23-s − 18·24-s + 31·25-s + 36·26-s − 3·27-s + 24·28-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.577·3-s + 4·4-s + 2.68·5-s − 1.63·6-s + 1.13·7-s + 6.36·8-s − 2/3·9-s + 7.58·10-s − 2.41·11-s − 2.30·12-s + 2.49·13-s + 3.20·14-s − 1.54·15-s + 17/2·16-s − 1.69·17-s − 1.88·18-s + 1.14·19-s + 10.7·20-s − 0.654·21-s − 6.82·22-s − 3.75·23-s − 3.67·24-s + 31/5·25-s + 7.06·26-s − 0.577·27-s + 4.53·28-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 104.4368484104.4368484
L(12)L(\frac12) ≈\approx 104.4368484104.4368484
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+pT2−5T3+11T4−5pT5+p3T6−p4T7+p4T8)2 ( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2}
3 1+T+pT2+8T3+8T4−7T5+2pT6−56T7−137T8−56pT9+2p3T10−7p3T11+8p4T12+8p5T13+p7T14+p7T15+p8T16 1 + T + p T^{2} + 8 T^{3} + 8 T^{4} - 7 T^{5} + 2 p T^{6} - 56 T^{7} - 137 T^{8} - 56 p T^{9} + 2 p^{3} T^{10} - 7 p^{3} T^{11} + 8 p^{4} T^{12} + 8 p^{5} T^{13} + p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16}
5 (1−3T−2T2−3T3+51T4−3pT5−2p2T6−3p3T7+p4T8)2 ( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
7 1−3T+pT2−36T3+108T4−219T5+122pT6−2628T7+5483T8−2628pT9+122p3T10−219p3T11+108p4T12−36p5T13+p7T14−3p7T15+p8T16 1 - 3 T + p T^{2} - 36 T^{3} + 108 T^{4} - 219 T^{5} + 122 p T^{6} - 2628 T^{7} + 5483 T^{8} - 2628 p T^{9} + 122 p^{3} T^{10} - 219 p^{3} T^{11} + 108 p^{4} T^{12} - 36 p^{5} T^{13} + p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16}
11 1+8T+51T2+252T3+1196T4+5184T5+20789T6+76586T7+264267T8+76586pT9+20789p2T10+5184p3T11+1196p4T12+252p5T13+51p6T14+8p7T15+p8T16 1 + 8 T + 51 T^{2} + 252 T^{3} + 1196 T^{4} + 5184 T^{5} + 20789 T^{6} + 76586 T^{7} + 264267 T^{8} + 76586 p T^{9} + 20789 p^{2} T^{10} + 5184 p^{3} T^{11} + 1196 p^{4} T^{12} + 252 p^{5} T^{13} + 51 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16}
13 1−9T+58T2−177T3+378T4−252T5+452pT6−46026T7+235283T8−46026pT9+452p3T10−252p3T11+378p4T12−177p5T13+58p6T14−9p7T15+p8T16 1 - 9 T + 58 T^{2} - 177 T^{3} + 378 T^{4} - 252 T^{5} + 452 p T^{6} - 46026 T^{7} + 235283 T^{8} - 46026 p T^{9} + 452 p^{3} T^{10} - 252 p^{3} T^{11} + 378 p^{4} T^{12} - 177 p^{5} T^{13} + 58 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16}
17 1+7T+47T2+144T3+628T4+1131T5+13454T6+53842T7+353533T8+53842pT9+13454p2T10+1131p3T11+628p4T12+144p5T13+47p6T14+7p7T15+p8T16 1 + 7 T + 47 T^{2} + 144 T^{3} + 628 T^{4} + 1131 T^{5} + 13454 T^{6} + 53842 T^{7} + 353533 T^{8} + 53842 p T^{9} + 13454 p^{2} T^{10} + 1131 p^{3} T^{11} + 628 p^{4} T^{12} + 144 p^{5} T^{13} + 47 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16}
19 1−5T+pT2−160T3+800T4−2765T5+914pT6−88480T7+312079T8−88480pT9+914p3T10−2765p3T11+800p4T12−160p5T13+p7T14−5p7T15+p8T16 1 - 5 T + p T^{2} - 160 T^{3} + 800 T^{4} - 2765 T^{5} + 914 p T^{6} - 88480 T^{7} + 312079 T^{8} - 88480 p T^{9} + 914 p^{3} T^{10} - 2765 p^{3} T^{11} + 800 p^{4} T^{12} - 160 p^{5} T^{13} + p^{7} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16}
23 (1+9T+38T2+255T3+1741T4+255pT5+38p2T6+9p3T7+p4T8)2 ( 1 + 9 T + 38 T^{2} + 255 T^{3} + 1741 T^{4} + 255 p T^{5} + 38 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1−10T+31T2+160T3−2079T4+160pT5+31p2T6−10p3T7+p4T8)2 ( 1 - 10 T + 31 T^{2} + 160 T^{3} - 2079 T^{4} + 160 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
37 (1−4T−57T2+4T3+3368T4+4pT5−57p2T6−4p3T7+p4T8)2 ( 1 - 4 T - 57 T^{2} + 4 T^{3} + 3368 T^{4} + 4 p T^{5} - 57 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
41 1−12T+121T2−348T3+336T4+17304T5+1999pT6−1336404T7+16471727T8−1336404pT9+1999p3T10+17304p3T11+336p4T12−348p5T13+121p6T14−12p7T15+p8T16 1 - 12 T + 121 T^{2} - 348 T^{3} + 336 T^{4} + 17304 T^{5} + 1999 p T^{6} - 1336404 T^{7} + 16471727 T^{8} - 1336404 p T^{9} + 1999 p^{3} T^{10} + 17304 p^{3} T^{11} + 336 p^{4} T^{12} - 348 p^{5} T^{13} + 121 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16}
43 1+6T+63T2+418T3+2778T4−932T5+50731T6−406266T7−2563037T8−406266pT9+50731p2T10−932p3T11+2778p4T12+418p5T13+63p6T14+6p7T15+p8T16 1 + 6 T + 63 T^{2} + 418 T^{3} + 2778 T^{4} - 932 T^{5} + 50731 T^{6} - 406266 T^{7} - 2563037 T^{8} - 406266 p T^{9} + 50731 p^{2} T^{10} - 932 p^{3} T^{11} + 2778 p^{4} T^{12} + 418 p^{5} T^{13} + 63 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16}
47 (1−2T−23T2+290T3+831T4+290pT5−23p2T6−2p3T7+p4T8)2 ( 1 - 2 T - 23 T^{2} + 290 T^{3} + 831 T^{4} + 290 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 1−9T−37T2+588T3−2592T4+22563T5−122714T6−1504026T7+23688233T8−1504026pT9−122714p2T10+22563p3T11−2592p4T12+588p5T13−37p6T14−9p7T15+p8T16 1 - 9 T - 37 T^{2} + 588 T^{3} - 2592 T^{4} + 22563 T^{5} - 122714 T^{6} - 1504026 T^{7} + 23688233 T^{8} - 1504026 p T^{9} - 122714 p^{2} T^{10} + 22563 p^{3} T^{11} - 2592 p^{4} T^{12} + 588 p^{5} T^{13} - 37 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16}
59 1−15T+199T2−1620T3+15540T4−116895T5+1250126T6−9793080T7+87433079T8−9793080pT9+1250126p2T10−116895p3T11+15540p4T12−1620p5T13+199p6T14−15p7T15+p8T16 1 - 15 T + 199 T^{2} - 1620 T^{3} + 15540 T^{4} - 116895 T^{5} + 1250126 T^{6} - 9793080 T^{7} + 87433079 T^{8} - 9793080 p T^{9} + 1250126 p^{2} T^{10} - 116895 p^{3} T^{11} + 15540 p^{4} T^{12} - 1620 p^{5} T^{13} + 199 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16}
61 (1−4T+46T2−4pT3+p2T4)4 ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4}
67 (1−4T−117T2+4T3+12128T4+4pT5−117p2T6−4p3T7+p4T8)2 ( 1 - 4 T - 117 T^{2} + 4 T^{3} + 12128 T^{4} + 4 p T^{5} - 117 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}
71 1+18T+271T2+2502T3+25686T4+226044T5+2596679T6+24000066T7+229833107T8+24000066pT9+2596679p2T10+226044p3T11+25686p4T12+2502p5T13+271p6T14+18p7T15+p8T16 1 + 18 T + 271 T^{2} + 2502 T^{3} + 25686 T^{4} + 226044 T^{5} + 2596679 T^{6} + 24000066 T^{7} + 229833107 T^{8} + 24000066 p T^{9} + 2596679 p^{2} T^{10} + 226044 p^{3} T^{11} + 25686 p^{4} T^{12} + 2502 p^{5} T^{13} + 271 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16}
73 1−24T+343T2−1872T3−7452T4+251688T5−1300339T6−7670556T7+170336663T8−7670556pT9−1300339p2T10+251688p3T11−7452p4T12−1872p5T13+343p6T14−24p7T15+p8T16 1 - 24 T + 343 T^{2} - 1872 T^{3} - 7452 T^{4} + 251688 T^{5} - 1300339 T^{6} - 7670556 T^{7} + 170336663 T^{8} - 7670556 p T^{9} - 1300339 p^{2} T^{10} + 251688 p^{3} T^{11} - 7452 p^{4} T^{12} - 1872 p^{5} T^{13} + 343 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16}
79 1+pT2−p3T6−p4T8−p5T10+p7T14+p8T16 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16}
83 1+11T+108T2−1167T3−19912T4−220392T5−157264T6+14295074T7+219932193T8+14295074pT9−157264p2T10−220392p3T11−19912p4T12−1167p5T13+108p6T14+11p7T15+p8T16 1 + 11 T + 108 T^{2} - 1167 T^{3} - 19912 T^{4} - 220392 T^{5} - 157264 T^{6} + 14295074 T^{7} + 219932193 T^{8} + 14295074 p T^{9} - 157264 p^{2} T^{10} - 220392 p^{3} T^{11} - 19912 p^{4} T^{12} - 1167 p^{5} T^{13} + 108 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16}
89 (1−10T−29T2+1060T3−7299T4+1060pT5−29p2T6−10p3T7+p4T8)2 ( 1 - 10 T - 29 T^{2} + 1060 T^{3} - 7299 T^{4} + 1060 p T^{5} - 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
97 (1−27T+182T2+2625T3−53249T4+2625pT5+182p2T6−27p3T7+p4T8)2 ( 1 - 27 T + 182 T^{2} + 2625 T^{3} - 53249 T^{4} + 2625 p T^{5} + 182 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2}
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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

−4.47564753414944653799599259156, −4.35968728975303028852142546871, −3.90991133505330854836096114206, −3.87902653562053664384338689258, −3.85363529720630644249830907077, −3.84187127821276164649644786489, −3.61321018264443534191077253947, −3.49448661823778809456751288551, −3.47383800564050540754393063253, −3.10298284152902397063568444360, −2.73136970837078241374868833766, −2.68151284045379050695067268689, −2.54390933214619382194819548639, −2.43243511668711924672276153631, −2.33158367423491637773942860434, −2.31277619470865572329702827547, −2.26810333519197052247130878578, −1.97708343660981188382211632110, −1.82131486956463051310435779230, −1.53334860072137652060278531357, −1.37809781561638034671461699036, −1.18893176681751076926950158809, −0.987628108808365433353935412980, −0.75514851786273606817133803720, −0.46677001763734334004821129215, 0.46677001763734334004821129215, 0.75514851786273606817133803720, 0.987628108808365433353935412980, 1.18893176681751076926950158809, 1.37809781561638034671461699036, 1.53334860072137652060278531357, 1.82131486956463051310435779230, 1.97708343660981188382211632110, 2.26810333519197052247130878578, 2.31277619470865572329702827547, 2.33158367423491637773942860434, 2.43243511668711924672276153631, 2.54390933214619382194819548639, 2.68151284045379050695067268689, 2.73136970837078241374868833766, 3.10298284152902397063568444360, 3.47383800564050540754393063253, 3.49448661823778809456751288551, 3.61321018264443534191077253947, 3.84187127821276164649644786489, 3.85363529720630644249830907077, 3.87902653562053664384338689258, 3.90991133505330854836096114206, 4.35968728975303028852142546871, 4.47564753414944653799599259156

Graph of the ZZ-function along the critical line

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