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Properties

Label 2-931-133.132-c1-0-23
Degree $2$
Conductor $931$
Sign $-0.765 + 0.643i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.27i·2-s − 2.74·3-s − 3.17·4-s − 1.16i·5-s + 6.23i·6-s + 2.67i·8-s + 4.51·9-s − 2.66·10-s + 4.83·11-s + 8.71·12-s + 3.16·13-s + 3.20i·15-s − 0.258·16-s + 3.82i·17-s − 10.2i·18-s + (3.29 + 2.85i)19-s + ⋯
L(s)  = 1  − 1.60i·2-s − 1.58·3-s − 1.58·4-s − 0.522i·5-s + 2.54i·6-s + 0.947i·8-s + 1.50·9-s − 0.841·10-s + 1.45·11-s + 2.51·12-s + 0.876·13-s + 0.827i·15-s − 0.0646·16-s + 0.928i·17-s − 2.42i·18-s + (0.755 + 0.655i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.765 + 0.643i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337639 - 0.926711i\)
\(L(\frac12)\) \(\approx\) \(0.337639 - 0.926711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (-3.29 - 2.85i)T \)
good2 \( 1 + 2.27iT - 2T^{2} \)
3 \( 1 + 2.74T + 3T^{2} \)
5 \( 1 + 1.16iT - 5T^{2} \)
11 \( 1 - 4.83T + 11T^{2} \)
13 \( 1 - 3.16T + 13T^{2} \)
17 \( 1 - 3.82iT - 17T^{2} \)
23 \( 1 - 7.97T + 23T^{2} \)
29 \( 1 + 7.81iT - 29T^{2} \)
31 \( 1 + 3.32T + 31T^{2} \)
37 \( 1 - 2.56iT - 37T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 + 6.25T + 43T^{2} \)
47 \( 1 - 1.49iT - 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 - 4.81iT - 67T^{2} \)
71 \( 1 - 9.62iT - 71T^{2} \)
73 \( 1 + 3.07iT - 73T^{2} \)
79 \( 1 + 4.04iT - 79T^{2} \)
83 \( 1 - 4.15iT - 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 0.0766T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08160552338541095764512271241, −9.264986600510291190894697830164, −8.486055225037582965011429148058, −6.88713049087731419936529992295, −6.12052932553022781479356919255, −5.12105300169728870847513822774, −4.24463323251531537475158141886, −3.42564289404794727910628843320, −1.52948652314129802105920458073, −0.901567302200547353282381105898, 0.976817478675307862118851707455, 3.49879636703802260543668237189, 4.88606702031767309302466269445, 5.28465960288143425798155305556, 6.32729521154951815518790944856, 6.92772136564168381104500453951, 7.14735745080274311588040547016, 8.744837793669778431790649697046, 9.232425776212565033827012498199, 10.48067209045040009218235716130

Graph of the $Z$-function along the critical line