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Properties

Label 2-31-31.30-c18-0-20
Degree 22
Conductor 3131
Sign 11
Analytic cond. 63.669763.6697
Root an. cond. 7.979327.97932
Motivic weight 1818
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 495·2-s − 1.71e4·4-s − 3.35e6·5-s + 7.22e7·7-s + 1.38e8·8-s + 3.87e8·9-s + 1.66e9·10-s − 3.57e10·14-s − 6.39e10·16-s − 1.91e11·18-s − 3.01e11·19-s + 5.74e10·20-s + 7.44e12·25-s − 1.23e12·28-s − 2.64e13·31-s − 4.58e12·32-s − 2.42e14·35-s − 6.63e12·36-s + 1.49e14·38-s − 4.63e14·40-s + 6.40e14·41-s − 1.30e15·45-s − 4.84e14·47-s + 3.59e15·49-s − 3.68e15·50-s + 9.98e15·56-s − 1.47e16·59-s + ⋯
L(s)  = 1  − 0.966·2-s − 0.0653·4-s − 1.71·5-s + 1.79·7-s + 1.02·8-s + 9-s + 1.66·10-s − 1.73·14-s − 0.930·16-s − 0.966·18-s − 0.934·19-s + 0.112·20-s + 1.95·25-s − 0.116·28-s − 31-s − 0.130·32-s − 3.07·35-s − 0.0653·36-s + 0.903·38-s − 1.76·40-s + 1.95·41-s − 1.71·45-s − 0.432·47-s + 2.20·49-s − 1.88·50-s + 1.84·56-s − 1.70·59-s + ⋯

Functional equation

Λ(s)=(31s/2 ΓC(s) L(s)=( Λ(19−s)\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}
Λ(s)=(31s/2 ΓC(s+9) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3131
Sign: 11
Analytic conductor: 63.669763.6697
Root analytic conductor: 7.979327.97932
Motivic weight: 1818
Rational: yes
Arithmetic: yes
Character: χ31(30,⋅)\chi_{31} (30, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 31, ( :9), 1)(2,\ 31,\ (\ :9),\ 1)

Particular Values

L(192)L(\frac{19}{2}) ≈\approx 0.93259143050.9325914305
L(12)L(\frac12) ≈\approx 0.93259143050.9325914305
L(10)L(10) 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+p9T 1 + p^{9} T
good2 1+495T+p18T2 1 + 495 T + p^{18} T^{2}
3 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
5 1+3355686T+p18T2 1 + 3355686 T + p^{18} T^{2}
7 1−72260210T+p18T2 1 - 72260210 T + p^{18} T^{2}
11 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
13 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
17 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
19 1+301505744342T+p18T2 1 + 301505744342 T + p^{18} T^{2}
23 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
29 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
37 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
41 1−640887818618322T+p18T2 1 - 640887818618322 T + p^{18} T^{2}
43 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
47 1+484327216285470T+p18T2 1 + 484327216285470 T + p^{18} T^{2}
53 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
59 1+14753739003245478T+p18T2 1 + 14753739003245478 T + p^{18} T^{2}
61 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
67 1−33886344531727690T+p18T2 1 - 33886344531727690 T + p^{18} T^{2}
71 1+86260446147898062T+p18T2 1 + 86260446147898062 T + p^{18} T^{2}
73 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
79 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
83 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
89 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
97 1−1388340453162394370T+p18T2 1 - 1388340453162394370 T + p^{18} 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

−12.67845650278808703327645762466, −11.35854022486146393672938933941, −10.63610399073561460039451573643, −8.885730664245830204405796122131, −7.910012607207070659030372306191, −7.38970743570324276010840878336, −4.69704901047632568819861724811, −4.09507367208330216107926038283, −1.71411407568811048568923496985, −0.63524853940931230639204006767, 0.63524853940931230639204006767, 1.71411407568811048568923496985, 4.09507367208330216107926038283, 4.69704901047632568819861724811, 7.38970743570324276010840878336, 7.910012607207070659030372306191, 8.885730664245830204405796122131, 10.63610399073561460039451573643, 11.35854022486146393672938933941, 12.67845650278808703327645762466

Graph of the ZZ-function along the critical line