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Properties

Label 2-31-31.30-c12-0-23
Degree 22
Conductor 3131
Sign 11
Analytic cond. 28.333828.3338
Root an. cond. 5.322955.32295
Motivic weight 1212
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 97·2-s + 5.31e3·4-s + 2.92e4·5-s − 5.03e4·7-s + 1.18e5·8-s + 5.31e5·9-s + 2.83e6·10-s − 4.88e6·14-s − 1.03e7·16-s + 5.15e7·18-s + 1.86e7·19-s + 1.55e8·20-s + 6.12e8·25-s − 2.67e8·28-s + 8.87e8·31-s − 1.48e9·32-s − 1.47e9·35-s + 2.82e9·36-s + 1.80e9·38-s + 3.45e9·40-s − 5.83e9·41-s + 1.55e10·45-s + 7.96e9·47-s − 1.13e10·49-s + 5.93e10·50-s − 5.94e9·56-s − 6.56e10·59-s + ⋯
L(s)  = 1  + 1.51·2-s + 1.29·4-s + 1.87·5-s − 0.428·7-s + 0.450·8-s + 9-s + 2.83·10-s − 0.649·14-s − 0.614·16-s + 1.51·18-s + 0.396·19-s + 2.42·20-s + 2.50·25-s − 0.555·28-s + 31-s − 1.38·32-s − 0.802·35-s + 1.29·36-s + 0.600·38-s + 0.843·40-s − 1.22·41-s + 1.87·45-s + 0.738·47-s − 0.816·49-s + 3.80·50-s − 0.192·56-s − 1.55·59-s + ⋯

Functional equation

Λ(s)=(31s/2 ΓC(s) L(s)=( Λ(13−s)\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}
Λ(s)=(31s/2 ΓC(s+6) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3131
Sign: 11
Analytic conductor: 28.333828.3338
Root analytic conductor: 5.322955.32295
Motivic weight: 1212
Rational: yes
Arithmetic: yes
Character: χ31(30,⋅)\chi_{31} (30, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 31, ( :6), 1)(2,\ 31,\ (\ :6),\ 1)

Particular Values

L(132)L(\frac{13}{2}) ≈\approx 6.1995659846.199565984
L(12)L(\frac12) ≈\approx 6.1995659846.199565984
L(7)L(7) 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−p6T 1 - p^{6} T
good2 1−97T+p12T2 1 - 97 T + p^{12} T^{2}
3 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
5 1−29266T+p12T2 1 - 29266 T + p^{12} T^{2}
7 1+50398T+p12T2 1 + 50398 T + p^{12} T^{2}
11 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
13 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
17 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
19 1−18650162T+p12T2 1 - 18650162 T + p^{12} T^{2}
23 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
29 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
37 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
41 1+5832937118T+p12T2 1 + 5832937118 T + p^{12} T^{2}
43 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
47 1−7960245442T+p12T2 1 - 7960245442 T + p^{12} T^{2}
53 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
59 1+65635334318T+p12T2 1 + 65635334318 T + p^{12} T^{2}
61 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
67 1+163049095438T+p12T2 1 + 163049095438 T + p^{12} T^{2}
71 1+175443432158T+p12T2 1 + 175443432158 T + p^{12} T^{2}
73 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
79 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
83 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
89 (1−p6T)(1+p6T) ( 1 - p^{6} T )( 1 + p^{6} T )
97 1−1601076090242T+p12T2 1 - 1601076090242 T + p^{12} 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

−13.75328513806506796312709347707, −13.29480697601699309174747576576, −12.25213639546959925837410473330, −10.40047624323426198151318492465, −9.354213755740272463729800126823, −6.78450523155803507121879541465, −5.85126863394289819991167382666, −4.67670584451918799826720606738, −2.95255408877476123530271839540, −1.62150884721752149568212367164, 1.62150884721752149568212367164, 2.95255408877476123530271839540, 4.67670584451918799826720606738, 5.85126863394289819991167382666, 6.78450523155803507121879541465, 9.354213755740272463729800126823, 10.40047624323426198151318492465, 12.25213639546959925837410473330, 13.29480697601699309174747576576, 13.75328513806506796312709347707

Graph of the ZZ-function along the critical line