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Properties

Label 2-931-133.130-c1-0-10
Degree $2$
Conductor $931$
Sign $-0.385 + 0.922i$
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  + (−0.439 + 2.49i)2-s + (−0.113 + 0.642i)3-s + (−4.14 − 1.50i)4-s + (1.26 − 0.460i)5-s + (−1.55 − 0.565i)6-s + (3.05 − 5.28i)8-s + (2.41 + 0.880i)9-s + (0.592 + 3.35i)10-s + (0.592 + 1.02i)11-s + (1.43 − 2.49i)12-s + (−2.55 − 0.929i)13-s + (0.152 + 0.866i)15-s + (5.08 + 4.26i)16-s + (−3.64 + 1.32i)17-s + (−3.25 + 5.64i)18-s + (−4.11 + 1.43i)19-s + ⋯
L(s)  = 1  + (−0.310 + 1.76i)2-s + (−0.0654 + 0.371i)3-s + (−2.07 − 0.754i)4-s + (0.566 − 0.206i)5-s + (−0.634 − 0.230i)6-s + (1.07 − 1.86i)8-s + (0.806 + 0.293i)9-s + (0.187 + 1.06i)10-s + (0.178 + 0.309i)11-s + (0.415 − 0.719i)12-s + (−0.708 − 0.257i)13-s + (0.0394 + 0.223i)15-s + (1.27 + 1.06i)16-s + (−0.884 + 0.321i)17-s + (−0.768 + 1.33i)18-s + (−0.944 + 0.328i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.385 + 0.922i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410707 - 0.616999i\)
\(L(\frac12)\) \(\approx\) \(0.410707 - 0.616999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.11 - 1.43i)T \)
good2 \( 1 + (0.439 - 2.49i)T + (-1.87 - 0.684i)T^{2} \)
3 \( 1 + (0.113 - 0.642i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-1.26 + 0.460i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-0.592 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.55 + 0.929i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.64 - 1.32i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (3.87 - 3.25i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.56 - 2.99i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + 3.83T + 31T^{2} \)
37 \( 1 + (-2.05 - 3.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.38 + 3.41i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.51 - 8.57i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.539 + 0.196i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.76 + 1.00i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-3.69 + 1.34i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.46 - 2.90i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.674 + 3.82i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (1.20 - 6.83i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.06 + 6.03i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-7.51 - 6.30i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.15 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.421 + 2.38i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-5.64 - 4.73i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.13626652001707786321383174922, −9.572972937061252885900100740642, −8.944053345113761204955776847739, −7.88704346833413592044049998291, −7.31486640318426352196403648917, −6.38147021746006678990398465560, −5.63639933820782513351887509902, −4.76224342765053481316921125936, −4.02528548521930849192568521689, −1.85097575677684125571095232983, 0.37838179817233165937486789548, 1.93742995674283923658214964682, 2.43633505136514477321800896601, 3.91930155285866054091947484615, 4.53161314307450842654593699136, 6.00636426701366167647741414650, 6.99114015384536020018404868332, 8.107338758104081734821350398181, 9.145398560050002449228586723133, 9.612727528928966880304976186199

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