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Properties

Label 2-13-13.12-c11-0-11
Degree $2$
Conductor $13$
Sign $0.845 - 0.534i$
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  − 85.7i·2-s − 584.·3-s − 5.30e3·4-s − 6.97e3i·5-s + 5.01e4i·6-s − 4.11e4i·7-s + 2.79e5i·8-s + 1.64e5·9-s − 5.98e5·10-s − 5.02e5i·11-s + 3.10e6·12-s + (1.13e6 − 7.14e5i)13-s − 3.52e6·14-s + 4.07e6i·15-s + 1.30e7·16-s − 7.46e6·17-s + ⋯
L(s)  = 1  − 1.89i·2-s − 1.38·3-s − 2.58·4-s − 0.998i·5-s + 2.63i·6-s − 0.925i·7-s + 3.01i·8-s + 0.929·9-s − 1.89·10-s − 0.940i·11-s + 3.59·12-s + (0.845 − 0.534i)13-s − 1.75·14-s + 1.38i·15-s + 3.11·16-s − 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.845 - 0.534i$
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: \((2,\ 13,\ (\ :11/2),\ 0.845 - 0.534i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.368252 + 0.106566i\)
\(L(\frac12)\) \(\approx\) \(0.368252 + 0.106566i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.13e6 + 7.14e5i)T \)
good2 \( 1 + 85.7iT - 2.04e3T^{2} \)
3 \( 1 + 584.T + 1.77e5T^{2} \)
5 \( 1 + 6.97e3iT - 4.88e7T^{2} \)
7 \( 1 + 4.11e4iT - 1.97e9T^{2} \)
11 \( 1 + 5.02e5iT - 2.85e11T^{2} \)
17 \( 1 + 7.46e6T + 3.42e13T^{2} \)
19 \( 1 - 9.86e6iT - 1.16e14T^{2} \)
23 \( 1 - 3.09e7T + 9.52e14T^{2} \)
29 \( 1 + 1.95e8T + 1.22e16T^{2} \)
31 \( 1 - 1.15e7iT - 2.54e16T^{2} \)
37 \( 1 + 1.48e8iT - 1.77e17T^{2} \)
41 \( 1 + 9.79e8iT - 5.50e17T^{2} \)
43 \( 1 + 6.27e6T + 9.29e17T^{2} \)
47 \( 1 + 3.75e8iT - 2.47e18T^{2} \)
53 \( 1 + 9.10e8T + 9.26e18T^{2} \)
59 \( 1 - 1.75e9iT - 3.01e19T^{2} \)
61 \( 1 + 8.61e8T + 4.35e19T^{2} \)
67 \( 1 - 1.06e10iT - 1.22e20T^{2} \)
71 \( 1 - 3.89e9iT - 2.31e20T^{2} \)
73 \( 1 + 6.08e9iT - 3.13e20T^{2} \)
79 \( 1 - 3.82e9T + 7.47e20T^{2} \)
83 \( 1 - 5.34e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.76e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.11e11iT - 7.15e21T^{2} \)
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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

−16.57084966815812569903071583276, −13.50524312969023103194356224488, −12.65366332414387541896148982050, −11.26074852269779155602364736692, −10.64188978725232294668105031114, −8.832692229283474734514841244658, −5.40588197496167221487712266861, −3.94382801205085706737539642798, −1.16203795819912412654467841795, −0.26527247230906066964892697605, 4.81273302653871232583816651045, 6.19885287863374774476064465374, 7.00507515601396873195782722526, 9.127888590497685643718072463167, 11.17592173188506415149496801270, 13.09159021847395054138904811089, 14.88543530597204926143627245563, 15.65025664135974362018716041245, 16.97191870214147530709976171759, 17.98832077589480832146131076433

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