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Properties

Label 2-31-31.25-c1-0-1
Degree $2$
Conductor $31$
Sign $0.998 + 0.0543i$
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  + 0.414·2-s + (−0.207 − 0.358i)3-s − 1.82·4-s + (−0.5 + 0.866i)5-s + (−0.0857 − 0.148i)6-s + (0.207 + 0.358i)7-s − 1.58·8-s + (1.41 − 2.44i)9-s + (−0.207 + 0.358i)10-s + (−1.62 + 2.80i)11-s + (0.378 + 0.655i)12-s + (1.91 − 3.31i)13-s + (0.0857 + 0.148i)14-s + 0.414·15-s + 3·16-s + (2.91 + 5.04i)17-s + ⋯
L(s)  = 1  + 0.292·2-s + (−0.119 − 0.207i)3-s − 0.914·4-s + (−0.223 + 0.387i)5-s + (−0.0350 − 0.0606i)6-s + (0.0782 + 0.135i)7-s − 0.560·8-s + (0.471 − 0.816i)9-s + (−0.0654 + 0.113i)10-s + (−0.488 + 0.846i)11-s + (0.109 + 0.189i)12-s + (0.530 − 0.919i)13-s + (0.0229 + 0.0397i)14-s + 0.106·15-s + 0.750·16-s + (0.706 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.998 + 0.0543i$
Analytic conductor: \(0.247536\)
Root analytic conductor: \(0.497530\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1/2),\ 0.998 + 0.0543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674635 - 0.0183575i\)
\(L(\frac12)\) \(\approx\) \(0.674635 - 0.0183575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (5 + 2.44i)T \)
good2 \( 1 - 0.414T + 2T^{2} \)
3 \( 1 + (0.207 + 0.358i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.207 - 0.358i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.62 - 2.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.91 + 3.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.91 - 5.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.20 + 3.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.44 - 9.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.65T + 47T^{2} \)
53 \( 1 + (2.91 - 5.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.03 + 3.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + (1.62 - 2.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0355 - 0.0615i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.914 + 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.37 - 5.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.03 + 8.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.48T + 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−17.31862213490014316712620472764, −15.36814349422185225817191129035, −14.73664129852492283657761407743, −13.07778824150064539416153977458, −12.45864769530597376429176992395, −10.63266447894344353369695100852, −9.248566860357097575608400920380, −7.65023331655886740009580965906, −5.77236974894511654073662100791, −3.87027652297521950499404573403, 4.10575882823759228164850980824, 5.52882929282229684328658185625, 7.88257003994256340065368716256, 9.210521158862579387229930737215, 10.67766489343954453639141019555, 12.23484494524968300585607223052, 13.50823021561560571067342056870, 14.27611593585774244888713052754, 16.05018576631684118125182052888, 16.70827283432450506578868596694

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