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Properties

Label 2-13-13.8-c10-0-4
Degree $2$
Conductor $13$
Sign $0.432 + 0.901i$
Analytic cond. $8.25964$
Root an. cond. $2.87395$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−35.8 − 35.8i)2-s − 170.·3-s + 1.55e3i·4-s + (1.96e3 + 1.96e3i)5-s + (6.10e3 + 6.10e3i)6-s + (224. − 224. i)7-s + (1.89e4 − 1.89e4i)8-s − 3.01e4·9-s − 1.40e5i·10-s + (5.25e4 − 5.25e4i)11-s − 2.64e5i·12-s + (2.73e5 − 2.50e5i)13-s − 1.61e4·14-s + (−3.33e5 − 3.33e5i)15-s + 2.26e5·16-s + 3.58e4i·17-s + ⋯
L(s)  = 1  + (−1.12 − 1.12i)2-s − 0.699·3-s + 1.51i·4-s + (0.627 + 0.627i)5-s + (0.785 + 0.785i)6-s + (0.0133 − 0.0133i)7-s + (0.579 − 0.579i)8-s − 0.510·9-s − 1.40i·10-s + (0.326 − 0.326i)11-s − 1.06i·12-s + (0.737 − 0.674i)13-s − 0.0300·14-s + (−0.439 − 0.439i)15-s + 0.216·16-s + 0.0252i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.432 + 0.901i$
Analytic conductor: \(8.25964\)
Root analytic conductor: \(2.87395\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :5),\ 0.432 + 0.901i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.646751 - 0.407213i\)
\(L(\frac12)\) \(\approx\) \(0.646751 - 0.407213i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.73e5 + 2.50e5i)T \)
good2 \( 1 + (35.8 + 35.8i)T + 1.02e3iT^{2} \)
3 \( 1 + 170.T + 5.90e4T^{2} \)
5 \( 1 + (-1.96e3 - 1.96e3i)T + 9.76e6iT^{2} \)
7 \( 1 + (-224. + 224. i)T - 2.82e8iT^{2} \)
11 \( 1 + (-5.25e4 + 5.25e4i)T - 2.59e10iT^{2} \)
17 \( 1 - 3.58e4iT - 2.01e12T^{2} \)
19 \( 1 + (-2.11e6 - 2.11e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 6.08e6iT - 4.14e13T^{2} \)
29 \( 1 - 2.39e7T + 4.20e14T^{2} \)
31 \( 1 + (-2.13e7 - 2.13e7i)T + 8.19e14iT^{2} \)
37 \( 1 + (-5.72e7 + 5.72e7i)T - 4.80e15iT^{2} \)
41 \( 1 + (7.52e7 + 7.52e7i)T + 1.34e16iT^{2} \)
43 \( 1 - 2.37e8iT - 2.16e16T^{2} \)
47 \( 1 + (1.74e7 - 1.74e7i)T - 5.25e16iT^{2} \)
53 \( 1 - 6.13e8T + 1.74e17T^{2} \)
59 \( 1 + (-3.68e7 + 3.68e7i)T - 5.11e17iT^{2} \)
61 \( 1 + 2.89e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.62e9 - 1.62e9i)T + 1.82e18iT^{2} \)
71 \( 1 + (1.52e9 + 1.52e9i)T + 3.25e18iT^{2} \)
73 \( 1 + (-2.81e9 + 2.81e9i)T - 4.29e18iT^{2} \)
79 \( 1 + 9.72e8T + 9.46e18T^{2} \)
83 \( 1 + (2.25e9 + 2.25e9i)T + 1.55e19iT^{2} \)
89 \( 1 + (4.67e9 - 4.67e9i)T - 3.11e19iT^{2} \)
97 \( 1 + (6.66e9 + 6.66e9i)T + 7.37e19iT^{2} \)
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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

−17.74056081209831825751235038772, −16.44625090270928988349380462454, −14.17953400684155123285443756842, −12.22258003919546723781082970493, −11.00647931404433386054995264231, −10.10767824490716126010855993928, −8.437273375810334605974950205935, −6.05756414095911410563809611672, −2.88570399360075325870304225237, −0.903300082030041442278136163306, 0.969001884972988220995822999384, 5.40444560699079004986061101634, 6.70370923881957723264524645745, 8.588550919813203175007027205152, 9.743310717257547526189259837184, 11.63066128135628599760131408136, 13.71530448338579368833346127785, 15.48449643267999795013710729341, 16.71544530526582604728640432911, 17.35752006122777283643484481507

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