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Properties

Label 16-31e16-1.1-c1e8-0-7
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  + 12·2-s + 70·4-s + 4·7-s + 264·8-s + 6·9-s + 48·14-s + 729·16-s + 72·18-s − 4·19-s + 10·25-s + 280·28-s + 1.60e3·32-s + 420·36-s − 48·38-s − 12·41-s − 24·47-s + 34·49-s + 120·50-s + 1.05e3·56-s − 12·59-s + 24·63-s + 3.06e3·64-s − 24·67-s − 12·71-s + 1.58e3·72-s − 280·76-s + 22·81-s + ⋯
L(s)  = 1  + 8.48·2-s + 35·4-s + 1.51·7-s + 93.3·8-s + 2·9-s + 12.8·14-s + 182.·16-s + 16.9·18-s − 0.917·19-s + 2·25-s + 52.9·28-s + 284.·32-s + 70·36-s − 7.78·38-s − 1.87·41-s − 3.50·47-s + 34/7·49-s + 16.9·50-s + 141.·56-s − 1.56·59-s + 3.02·63-s + 383.·64-s − 2.93·67-s − 1.42·71-s + 186.·72-s − 32.1·76-s + 22/9·81-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 872.2199682872.2199682
L(12)L(\frac12) ≈\approx 872.2199682872.2199682
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 (1−3T+5T2−3pT3+p2T4)4 ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4}
3 1−2pT2+14T4−8pT6+79T8−8p3T10+14p4T12−2p7T14+p8T16 1 - 2 p T^{2} + 14 T^{4} - 8 p T^{6} + 79 T^{8} - 8 p^{3} T^{10} + 14 p^{4} T^{12} - 2 p^{7} T^{14} + p^{8} T^{16}
5 (1−pT2)4(1+pT2+p2T4)2 ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2}
7 (1−5T+pT2)4(1+4T+pT2)4 ( 1 - 5 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4}
11 (1−4T2−105T4−4p2T6+p4T8)2 ( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2}
13 1+2T2+70T4−808T6−24881T8−808p2T10+70p4T12+2p6T14+p8T16 1 + 2 T^{2} + 70 T^{4} - 808 T^{6} - 24881 T^{8} - 808 p^{2} T^{10} + 70 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16}
17 1−54T2+1654T4−36936T6+666399T8−36936p2T10+1654p4T12−54p6T14+p8T16 1 - 54 T^{2} + 1654 T^{4} - 36936 T^{6} + 666399 T^{8} - 36936 p^{2} T^{10} + 1654 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16}
19 (1−7T+pT2)4(1+8T+pT2)4 ( 1 - 7 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4}
23 (1+38T2+1014T4+38p2T6+p4T8)2 ( 1 + 38 T^{2} + 1014 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2}
29 (1+102T2+4238T4+102p2T6+p4T8)2 ( 1 + 102 T^{2} + 4238 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} )^{2}
37 (1−56T2+1767T4−56p2T6+p4T8)2 ( 1 - 56 T^{2} + 1767 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2}
41 (1+6T−35T2−66T3+1884T4−66pT5−35p2T6+6p3T7+p4T8)2 ( 1 + 6 T - 35 T^{2} - 66 T^{3} + 1884 T^{4} - 66 p T^{5} - 35 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 1−78T2+3070T4+53352T6−5583921T8+53352p2T10+3070p4T12−78p6T14+p8T16 1 - 78 T^{2} + 3070 T^{4} + 53352 T^{6} - 5583921 T^{8} + 53352 p^{2} T^{10} + 3070 p^{4} T^{12} - 78 p^{6} T^{14} + p^{8} T^{16}
47 (1+6T+58T2+6pT3+p2T4)4 ( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}
53 1+4T2+874T4−25904T6−7199261T8−25904p2T10+874p4T12+4p6T14+p8T16 1 + 4 T^{2} + 874 T^{4} - 25904 T^{6} - 7199261 T^{8} - 25904 p^{2} T^{10} + 874 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16}
59 (1+6T−11T2−426T3−3396T4−426pT5−11p2T6+6p3T7+p4T8)2 ( 1 + 6 T - 11 T^{2} - 426 T^{3} - 3396 T^{4} - 426 p T^{5} - 11 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
61 (1+30T2+62T4+30p2T6+p4T8)2 ( 1 + 30 T^{2} + 62 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2}
67 (1+6T−31T2+6pT3+p2T4)4 ( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}
71 (1+6T−95T2−66T3+9564T4−66pT5−95p2T6+6p3T7+p4T8)2 ( 1 + 6 T - 95 T^{2} - 66 T^{3} + 9564 T^{4} - 66 p T^{5} - 95 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 (1−128T2+11055T4−128p2T6+p4T8)2 ( 1 - 128 T^{2} + 11055 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2}
79 1−190T2+18238T4−1022200T6+54800863T8−1022200p2T10+18238p4T12−190p6T14+p8T16 1 - 190 T^{2} + 18238 T^{4} - 1022200 T^{6} + 54800863 T^{8} - 1022200 p^{2} T^{10} + 18238 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16}
83 (1−156T2+17447T4−156p2T6+p4T8)2 ( 1 - 156 T^{2} + 17447 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 (1+86T2+7566T4+86p2T6+p4T8)2 ( 1 + 86 T^{2} + 7566 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} )^{2}
97 (1+7T+pT2)8 ( 1 + 7 T + p T^{2} )^{8}
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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

−4.45616205812294243222859517793, −4.29239752928085699878730340213, −4.18573898230919691209262755907, −3.92017055504253646069073535816, −3.88430349662771912351977390043, −3.79629688923818590413286925721, −3.76236406528942971547124724460, −3.63720423363257536236207110281, −3.60031220386006664996698464801, −3.10340142104226912934817451413, −3.09020429810435355713124070890, −3.04956746895836336875379014082, −3.01279340836675752018292135730, −2.89043964156180262728687126617, −2.66471075011528838648739515231, −2.36369550693743985738672673935, −2.35094707746500028249797768808, −2.21264936014318618258699664624, −1.65059763698703105866690815670, −1.59923253466223188326047614869, −1.49297488491669257150666899869, −1.48445986689059585449791403409, −1.30001055532120730369052752256, −0.993279212698727776238547259557, −0.32377480651846747146300828799, 0.32377480651846747146300828799, 0.993279212698727776238547259557, 1.30001055532120730369052752256, 1.48445986689059585449791403409, 1.49297488491669257150666899869, 1.59923253466223188326047614869, 1.65059763698703105866690815670, 2.21264936014318618258699664624, 2.35094707746500028249797768808, 2.36369550693743985738672673935, 2.66471075011528838648739515231, 2.89043964156180262728687126617, 3.01279340836675752018292135730, 3.04956746895836336875379014082, 3.09020429810435355713124070890, 3.10340142104226912934817451413, 3.60031220386006664996698464801, 3.63720423363257536236207110281, 3.76236406528942971547124724460, 3.79629688923818590413286925721, 3.88430349662771912351977390043, 3.92017055504253646069073535816, 4.18573898230919691209262755907, 4.29239752928085699878730340213, 4.45616205812294243222859517793

Graph of the ZZ-function along the critical line

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