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Properties

Label 2-231-1.1-c3-0-13
Degree 22
Conductor 231231
Sign 11
Analytic cond. 13.629413.6294
Root an. cond. 3.691803.69180
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.561·2-s + 3·3-s − 7.68·4-s + 18.6·5-s + 1.68·6-s + 7·7-s − 8.80·8-s + 9·9-s + 10.4·10-s − 11·11-s − 23.0·12-s + 36.4·13-s + 3.93·14-s + 56.0·15-s + 56.5·16-s + 41.1·17-s + 5.05·18-s − 23.6·19-s − 143.·20-s + 21·21-s − 6.17·22-s − 140.·23-s − 26.4·24-s + 224.·25-s + 20.4·26-s + 27·27-s − 53.7·28-s + ⋯
L(s)  = 1  + 0.198·2-s + 0.577·3-s − 0.960·4-s + 1.67·5-s + 0.114·6-s + 0.377·7-s − 0.389·8-s + 0.333·9-s + 0.331·10-s − 0.301·11-s − 0.554·12-s + 0.777·13-s + 0.0750·14-s + 0.964·15-s + 0.883·16-s + 0.586·17-s + 0.0661·18-s − 0.286·19-s − 1.60·20-s + 0.218·21-s − 0.0598·22-s − 1.26·23-s − 0.224·24-s + 1.79·25-s + 0.154·26-s + 0.192·27-s − 0.363·28-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 13.629413.6294
Root analytic conductor: 3.691803.69180
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 231, ( :3/2), 1)(2,\ 231,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7080158572.708015857
L(12)L(\frac12) ≈\approx 2.7080158572.708015857
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 1−3T 1 - 3T
7 1−7T 1 - 7T
11 1+11T 1 + 11T
good2 1−0.561T+8T2 1 - 0.561T + 8T^{2}
5 1−18.6T+125T2 1 - 18.6T + 125T^{2}
13 1−36.4T+2.19e3T2 1 - 36.4T + 2.19e3T^{2}
17 1−41.1T+4.91e3T2 1 - 41.1T + 4.91e3T^{2}
19 1+23.6T+6.85e3T2 1 + 23.6T + 6.85e3T^{2}
23 1+140.T+1.21e4T2 1 + 140.T + 1.21e4T^{2}
29 1−278.T+2.43e4T2 1 - 278.T + 2.43e4T^{2}
31 1−191.T+2.97e4T2 1 - 191.T + 2.97e4T^{2}
37 1−196.T+5.06e4T2 1 - 196.T + 5.06e4T^{2}
41 1+322.T+6.89e4T2 1 + 322.T + 6.89e4T^{2}
43 1+3.67T+7.95e4T2 1 + 3.67T + 7.95e4T^{2}
47 1+397.T+1.03e5T2 1 + 397.T + 1.03e5T^{2}
53 1−597.T+1.48e5T2 1 - 597.T + 1.48e5T^{2}
59 1−668.T+2.05e5T2 1 - 668.T + 2.05e5T^{2}
61 1+667.T+2.26e5T2 1 + 667.T + 2.26e5T^{2}
67 1+730.T+3.00e5T2 1 + 730.T + 3.00e5T^{2}
71 1+31.2T+3.57e5T2 1 + 31.2T + 3.57e5T^{2}
73 1+434.T+3.89e5T2 1 + 434.T + 3.89e5T^{2}
79 1+782.T+4.93e5T2 1 + 782.T + 4.93e5T^{2}
83 1+426.T+5.71e5T2 1 + 426.T + 5.71e5T^{2}
89 1+899.T+7.04e5T2 1 + 899.T + 7.04e5T^{2}
97 1+942.T+9.12e5T2 1 + 942.T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.98919978424583024658441025743, −10.22086137157344132895095267758, −9.975513788543823733855051199593, −8.778654827260416413432319458108, −8.161433505919597216027701969169, −6.39369662888452053872925597664, −5.50172407959130252243984210451, −4.37035250923888642440931641446, −2.82463910437626107578622233504, −1.35609486101962426245410304782, 1.35609486101962426245410304782, 2.82463910437626107578622233504, 4.37035250923888642440931641446, 5.50172407959130252243984210451, 6.39369662888452053872925597664, 8.161433505919597216027701969169, 8.778654827260416413432319458108, 9.975513788543823733855051199593, 10.22086137157344132895095267758, 11.98919978424583024658441025743

Graph of the ZZ-function along the critical line