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Properties

Label 2-231-1.1-c3-0-10
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  − 3.63·2-s − 3·3-s + 5.18·4-s + 21.1·5-s + 10.8·6-s + 7·7-s + 10.2·8-s + 9·9-s − 76.6·10-s + 11·11-s − 15.5·12-s + 87.3·13-s − 25.4·14-s − 63.3·15-s − 78.6·16-s − 28.2·17-s − 32.6·18-s − 97.9·19-s + 109.·20-s − 21·21-s − 39.9·22-s − 112.·23-s − 30.6·24-s + 320.·25-s − 317.·26-s − 27·27-s + 36.2·28-s + ⋯
L(s)  = 1  − 1.28·2-s − 0.577·3-s + 0.647·4-s + 1.88·5-s + 0.741·6-s + 0.377·7-s + 0.452·8-s + 0.333·9-s − 2.42·10-s + 0.301·11-s − 0.373·12-s + 1.86·13-s − 0.485·14-s − 1.09·15-s − 1.22·16-s − 0.403·17-s − 0.427·18-s − 1.18·19-s + 1.22·20-s − 0.218·21-s − 0.387·22-s − 1.01·23-s − 0.261·24-s + 2.56·25-s − 2.39·26-s − 0.192·27-s + 0.244·28-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 1.1678352441.167835244
L(12)L(\frac12) ≈\approx 1.1678352441.167835244
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+3.63T+8T2 1 + 3.63T + 8T^{2}
5 1−21.1T+125T2 1 - 21.1T + 125T^{2}
13 1−87.3T+2.19e3T2 1 - 87.3T + 2.19e3T^{2}
17 1+28.2T+4.91e3T2 1 + 28.2T + 4.91e3T^{2}
19 1+97.9T+6.85e3T2 1 + 97.9T + 6.85e3T^{2}
23 1+112.T+1.21e4T2 1 + 112.T + 1.21e4T^{2}
29 1−14.9T+2.43e4T2 1 - 14.9T + 2.43e4T^{2}
31 1−138.T+2.97e4T2 1 - 138.T + 2.97e4T^{2}
37 1−206.T+5.06e4T2 1 - 206.T + 5.06e4T^{2}
41 1−321.T+6.89e4T2 1 - 321.T + 6.89e4T^{2}
43 1−285.T+7.95e4T2 1 - 285.T + 7.95e4T^{2}
47 1+303.T+1.03e5T2 1 + 303.T + 1.03e5T^{2}
53 1+554.T+1.48e5T2 1 + 554.T + 1.48e5T^{2}
59 1+693.T+2.05e5T2 1 + 693.T + 2.05e5T^{2}
61 1−156.T+2.26e5T2 1 - 156.T + 2.26e5T^{2}
67 1−584.T+3.00e5T2 1 - 584.T + 3.00e5T^{2}
71 1−363.T+3.57e5T2 1 - 363.T + 3.57e5T^{2}
73 1+747.T+3.89e5T2 1 + 747.T + 3.89e5T^{2}
79 1−419.T+4.93e5T2 1 - 419.T + 4.93e5T^{2}
83 1−1.17e3T+5.71e5T2 1 - 1.17e3T + 5.71e5T^{2}
89 1+397.T+7.04e5T2 1 + 397.T + 7.04e5T^{2}
97 1+1.33e3T+9.12e5T2 1 + 1.33e3T + 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

−11.12559454992077890728740463258, −10.68135051395737917094302759166, −9.742425860901052985000236764127, −9.001533901847018301292413133202, −8.118103597015872081320266812156, −6.45045136289045120851465561453, −6.03016613445236212322849883334, −4.48850284240629286807991395865, −2.05081071765436100564573873773, −1.09844256276300120927701175377, 1.09844256276300120927701175377, 2.05081071765436100564573873773, 4.48850284240629286807991395865, 6.03016613445236212322849883334, 6.45045136289045120851465561453, 8.118103597015872081320266812156, 9.001533901847018301292413133202, 9.742425860901052985000236764127, 10.68135051395737917094302759166, 11.12559454992077890728740463258

Graph of the ZZ-function along the critical line