Properties

Label 2-31-31.8-c1-0-0
Degree $2$
Conductor $31$
Sign $0.965 - 0.258i$
Analytic cond. $0.247536$
Root an. cond. $0.497530$
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  + (−0.190 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (1.30 + 0.951i)4-s − 2.61·5-s + 0.618·6-s + (−2.42 − 1.76i)7-s + (−1.80 + 1.31i)8-s + (1.61 − 1.17i)9-s + (0.5 − 1.53i)10-s + (0.618 + 0.449i)11-s + (0.499 − 1.53i)12-s + (1.5 + 4.61i)13-s + (1.5 − 1.08i)14-s + (0.809 + 2.48i)15-s + (0.572 + 1.76i)16-s + (−0.190 + 0.138i)17-s + ⋯
L(s)  = 1  + (−0.135 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.654 + 0.475i)4-s − 1.17·5-s + 0.252·6-s + (−0.917 − 0.666i)7-s + (−0.639 + 0.464i)8-s + (0.539 − 0.391i)9-s + (0.158 − 0.486i)10-s + (0.186 + 0.135i)11-s + (0.144 − 0.444i)12-s + (0.416 + 1.28i)13-s + (0.400 − 0.291i)14-s + (0.208 + 0.642i)15-s + (0.143 + 0.440i)16-s + (−0.0463 + 0.0336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(0.247536\)
Root analytic conductor: \(0.497530\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618700 + 0.0813922i\)
\(L(\frac12)\) \(\approx\) \(0.618700 + 0.0813922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (5.54 + 0.502i)T \)
good2 \( 1 + (0.190 - 0.587i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
7 \( 1 + (2.42 + 1.76i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.618 - 0.449i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.5 - 4.61i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.190 - 0.138i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.54 + 4.75i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.42 + 3.21i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.66 - 8.19i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 0.236T + 37T^{2} \)
41 \( 1 + (-2 + 6.15i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (1.42 - 4.39i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.04 + 3.21i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.2 + 7.46i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.92 - 9.00i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 6.94T + 61T^{2} \)
67 \( 1 + 4.23T + 67T^{2} \)
71 \( 1 + (0.0729 - 0.0530i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.92 - 5.03i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.26 - 3.88i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.16 - 3.75i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.28 - 3.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−16.68391637308476345249915296042, −16.08671870268119028061831331453, −14.97499019196540781530534036037, −13.14585306945360279976052428074, −12.10350767476457834285170163722, −11.08247801506541672351027421811, −8.989830554457769108652548569150, −7.19042644833126543563050771444, −6.82636523187518682935201420961, −3.76198785283307356381246442880, 3.46493639857579719593935929257, 5.78436243846332643293479433510, 7.62832989512876994316291626328, 9.546495903272986379630782905837, 10.68767631280942441817826275505, 11.76843876927165282474894896132, 12.91858372072629820218699870260, 15.18499768271482298977805682136, 15.60444874497047192092788843712, 16.49573578494699816380554537201

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