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Properties

Label 2-13-13.12-c11-0-4
Degree $2$
Conductor $13$
Sign $-0.886 - 0.462i$
Analytic cond. $9.98846$
Root an. cond. $3.16045$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 69.4i·2-s + 612.·3-s − 2.78e3·4-s + 159. i·5-s + 4.25e4i·6-s + 4.87e4i·7-s − 5.09e4i·8-s + 1.97e5·9-s − 1.10e4·10-s + 3.70e5i·11-s − 1.70e6·12-s + (−1.18e6 − 6.18e5i)13-s − 3.38e6·14-s + 9.75e4i·15-s − 2.15e6·16-s + 9.66e6·17-s + ⋯
L(s)  = 1  + 1.53i·2-s + 1.45·3-s − 1.35·4-s + 0.0227i·5-s + 2.23i·6-s + 1.09i·7-s − 0.549i·8-s + 1.11·9-s − 0.0350·10-s + 0.694i·11-s − 1.97·12-s + (−0.886 − 0.462i)13-s − 1.68·14-s + 0.0331i·15-s − 0.514·16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.886 - 0.462i$
Analytic conductor: \(9.98846\)
Root analytic conductor: \(3.16045\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :11/2),\ -0.886 - 0.462i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.600486 + 2.45086i\)
\(L(\frac12)\) \(\approx\) \(0.600486 + 2.45086i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.18e6 + 6.18e5i)T \)
good2 \( 1 - 69.4iT - 2.04e3T^{2} \)
3 \( 1 - 612.T + 1.77e5T^{2} \)
5 \( 1 - 159. iT - 4.88e7T^{2} \)
7 \( 1 - 4.87e4iT - 1.97e9T^{2} \)
11 \( 1 - 3.70e5iT - 2.85e11T^{2} \)
17 \( 1 - 9.66e6T + 3.42e13T^{2} \)
19 \( 1 + 1.21e7iT - 1.16e14T^{2} \)
23 \( 1 - 4.62e6T + 9.52e14T^{2} \)
29 \( 1 - 5.32e7T + 1.22e16T^{2} \)
31 \( 1 - 1.98e8iT - 2.54e16T^{2} \)
37 \( 1 + 4.05e8iT - 1.77e17T^{2} \)
41 \( 1 - 7.05e8iT - 5.50e17T^{2} \)
43 \( 1 - 1.44e9T + 9.29e17T^{2} \)
47 \( 1 + 2.05e9iT - 2.47e18T^{2} \)
53 \( 1 + 2.12e9T + 9.26e18T^{2} \)
59 \( 1 + 1.07e10iT - 3.01e19T^{2} \)
61 \( 1 - 1.42e9T + 4.35e19T^{2} \)
67 \( 1 - 2.65e9iT - 1.22e20T^{2} \)
71 \( 1 - 7.57e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.85e10iT - 3.13e20T^{2} \)
79 \( 1 - 7.69e9T + 7.47e20T^{2} \)
83 \( 1 + 3.97e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.42e10iT - 2.77e21T^{2} \)
97 \( 1 - 3.73e10iT - 7.15e21T^{2} \)
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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

−17.54559016100778587358387169465, −15.89367847602696324394206643478, −14.89069294993246611320994330208, −14.31980346626882810481569011389, −12.60233257478239382257978343602, −9.478779022165486762669684905102, −8.383379176538807209425222321510, −7.20653564266309335042708796775, −5.17543057126950321636765494228, −2.68563586813935483084516320969, 1.14179561561096450615493040451, 2.83209290426397028787857910885, 3.97572364138205912091681525490, 7.74936564157071523015186280072, 9.409981591346977129767236272356, 10.49314711338208298349480404025, 12.27145564794928563410412147660, 13.67594495255992570171487724503, 14.46781274313873947069020336614, 16.69671600638354496330492930306

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