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Properties

Label 2-931-133.132-c1-0-53
Degree $2$
Conductor $931$
Sign $-0.980 + 0.194i$
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  − 1.82i·2-s + 0.702·3-s − 1.34·4-s − 2.40i·5-s − 1.28i·6-s − 1.19i·8-s − 2.50·9-s − 4.39·10-s + 6.30·11-s − 0.945·12-s + 4.93·13-s − 1.68i·15-s − 4.88·16-s − 4.48i·17-s + 4.58i·18-s + (−4.24 − 0.975i)19-s + ⋯
L(s)  = 1  − 1.29i·2-s + 0.405·3-s − 0.672·4-s − 1.07i·5-s − 0.524i·6-s − 0.423i·8-s − 0.835·9-s − 1.38·10-s + 1.90·11-s − 0.272·12-s + 1.36·13-s − 0.435i·15-s − 1.22·16-s − 1.08i·17-s + 1.08i·18-s + (−0.974 − 0.223i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.980 + 0.194i$
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: \((2,\ 931,\ (\ :1/2),\ -0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185564 - 1.89038i\)
\(L(\frac12)\) \(\approx\) \(0.185564 - 1.89038i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.24 + 0.975i)T \)
good2 \( 1 + 1.82iT - 2T^{2} \)
3 \( 1 - 0.702T + 3T^{2} \)
5 \( 1 + 2.40iT - 5T^{2} \)
11 \( 1 - 6.30T + 11T^{2} \)
13 \( 1 - 4.93T + 13T^{2} \)
17 \( 1 + 4.48iT - 17T^{2} \)
23 \( 1 - 3.11T + 23T^{2} \)
29 \( 1 - 9.51iT - 29T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 + 5.58iT - 37T^{2} \)
41 \( 1 + 5.58T + 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 - 5.67iT - 47T^{2} \)
53 \( 1 + 1.58iT - 53T^{2} \)
59 \( 1 + 2.93T + 59T^{2} \)
61 \( 1 + 6.15iT - 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 + 4.82iT - 71T^{2} \)
73 \( 1 + 8.50iT - 73T^{2} \)
79 \( 1 - 7.30iT - 79T^{2} \)
83 \( 1 - 3.15iT - 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 5.33T + 97T^{2} \)
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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

−9.401795694252723902091602509281, −8.959978770020532878353206268581, −8.690882400157843462208170745069, −7.11069922252175927145760081410, −6.18861910310697819048432816055, −4.94945818179522382137847872070, −3.89218063804561869683607460120, −3.25174888981268937414192466147, −1.82955733173807044103944862581, −0.908898384458686113104546492629, 1.93677516512652299436145763645, 3.37460230240446107871606816199, 4.15965721150304285004533851006, 5.81941632633207467082349586601, 6.31204605744104244827981215039, 6.81871509065775739737741077746, 7.920206930424836523066870415373, 8.658850583582986221698755878906, 9.133289036799497388181835562260, 10.50688955880996966379276296026

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