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Properties

Label 2-31-31.30-c6-0-3
Degree 22
Conductor 3131
Sign 11
Analytic cond. 7.131677.13167
Root an. cond. 2.670512.67051
Motivic weight 66
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15·2-s + 161·4-s − 246·5-s − 430·7-s − 1.45e3·8-s + 729·9-s + 3.69e3·10-s + 6.45e3·14-s + 1.15e4·16-s − 1.09e4·18-s + 1.06e4·19-s − 3.96e4·20-s + 4.48e4·25-s − 6.92e4·28-s − 2.97e4·31-s − 7.96e4·32-s + 1.05e5·35-s + 1.17e5·36-s − 1.59e5·38-s + 3.57e5·40-s − 6.05e4·41-s − 1.79e5·45-s + 1.71e5·47-s + 6.72e4·49-s − 6.73e5·50-s + 6.25e5·56-s + 1.36e5·59-s + ⋯
L(s)  = 1  − 1.87·2-s + 2.51·4-s − 1.96·5-s − 1.25·7-s − 2.84·8-s + 9-s + 3.68·10-s + 2.35·14-s + 2.81·16-s − 1.87·18-s + 1.54·19-s − 4.95·20-s + 2.87·25-s − 3.15·28-s − 31-s − 2.43·32-s + 2.46·35-s + 2.51·36-s − 2.90·38-s + 5.59·40-s − 0.878·41-s − 1.96·45-s + 1.65·47-s + 0.571·49-s − 5.38·50-s + 3.56·56-s + 0.666·59-s + ⋯

Functional equation

Λ(s)=(31s/2 ΓC(s) L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(31s/2 ΓC(s+3) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3131
Sign: 11
Analytic conductor: 7.131677.13167
Root analytic conductor: 2.670512.67051
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: χ31(30,⋅)\chi_{31} (30, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 31, ( :3), 1)(2,\ 31,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.34841602470.3484160247
L(12)L(\frac12) ≈\approx 0.34841602470.3484160247
L(4)L(4) 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+p3T 1 + p^{3} T
good2 1+15T+p6T2 1 + 15 T + p^{6} T^{2}
3 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
5 1+246T+p6T2 1 + 246 T + p^{6} T^{2}
7 1+430T+p6T2 1 + 430 T + p^{6} T^{2}
11 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
13 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
17 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
19 1−10618T+p6T2 1 - 10618 T + p^{6} T^{2}
23 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
29 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
37 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
41 1+60558T+p6T2 1 + 60558 T + p^{6} T^{2}
43 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
47 1−171810T+p6T2 1 - 171810 T + p^{6} T^{2}
53 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
59 1−136842T+p6T2 1 - 136842 T + p^{6} T^{2}
61 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
67 1+133670T+p6T2 1 + 133670 T + p^{6} T^{2}
71 1−284178T+p6T2 1 - 284178 T + p^{6} T^{2}
73 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
79 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
83 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
89 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
97 1−1807490T+p6T2 1 - 1807490 T + p^{6} T^{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

−15.94111592159695384775215938261, −15.36165536989464381497875641152, −12.51334466709860948944121497719, −11.53053452625768437971591688568, −10.25959883495607747915136177012, −9.065118761165065536808628316593, −7.64827883225499167030293822542, −6.97167795634074739591920598872, −3.42259970738871566827389033475, −0.65987254774365550354591134882, 0.65987254774365550354591134882, 3.42259970738871566827389033475, 6.97167795634074739591920598872, 7.64827883225499167030293822542, 9.065118761165065536808628316593, 10.25959883495607747915136177012, 11.53053452625768437971591688568, 12.51334466709860948944121497719, 15.36165536989464381497875641152, 15.94111592159695384775215938261

Graph of the ZZ-function along the critical line