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Properties

Label 2-13-13.12-c7-0-0
Degree $2$
Conductor $13$
Sign $-0.825 + 0.564i$
Analytic cond. $4.06100$
Root an. cond. $2.01519$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18.4i·2-s − 27.4·3-s − 213.·4-s + 70.5i·5-s − 506. i·6-s − 454. i·7-s − 1.58e3i·8-s − 1.43e3·9-s − 1.30e3·10-s + 6.20e3i·11-s + 5.86e3·12-s + (−6.53e3 + 4.47e3i)13-s + 8.41e3·14-s − 1.93e3i·15-s + 1.98e3·16-s + 2.79e4·17-s + ⋯
L(s)  = 1  + 1.63i·2-s − 0.586·3-s − 1.67·4-s + 0.252i·5-s − 0.958i·6-s − 0.501i·7-s − 1.09i·8-s − 0.656·9-s − 0.412·10-s + 1.40i·11-s + 0.979·12-s + (−0.825 + 0.564i)13-s + 0.819·14-s − 0.147i·15-s + 0.121·16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.825 + 0.564i$
Analytic conductor: \(4.06100\)
Root analytic conductor: \(2.01519\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :7/2),\ -0.825 + 0.564i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.220684 - 0.713781i\)
\(L(\frac12)\) \(\approx\) \(0.220684 - 0.713781i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (6.53e3 - 4.47e3i)T \)
good2 \( 1 - 18.4iT - 128T^{2} \)
3 \( 1 + 27.4T + 2.18e3T^{2} \)
5 \( 1 - 70.5iT - 7.81e4T^{2} \)
7 \( 1 + 454. iT - 8.23e5T^{2} \)
11 \( 1 - 6.20e3iT - 1.94e7T^{2} \)
17 \( 1 - 2.79e4T + 4.10e8T^{2} \)
19 \( 1 - 1.99e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.59e4T + 3.40e9T^{2} \)
29 \( 1 - 6.64e4T + 1.72e10T^{2} \)
31 \( 1 + 1.28e5iT - 2.75e10T^{2} \)
37 \( 1 - 6.08e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.21e4iT - 1.94e11T^{2} \)
43 \( 1 + 6.23e5T + 2.71e11T^{2} \)
47 \( 1 - 3.42e5iT - 5.06e11T^{2} \)
53 \( 1 + 6.76e5T + 1.17e12T^{2} \)
59 \( 1 + 6.00e5iT - 2.48e12T^{2} \)
61 \( 1 + 3.99e4T + 3.14e12T^{2} \)
67 \( 1 + 3.50e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.53e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.35e6iT - 1.10e13T^{2} \)
79 \( 1 - 1.39e6T + 1.92e13T^{2} \)
83 \( 1 - 7.87e6iT - 2.71e13T^{2} \)
89 \( 1 + 6.81e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.63e7iT - 8.07e13T^{2} \)
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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

−18.43437629283682764684629280618, −17.17364362267071899059927568475, −16.63457212241585093446789187304, −14.98420656366947987536918471886, −14.12246749854664479115891391072, −12.10287034628628273701706711224, −9.911969504320563643371150670113, −7.84038819283196400533718859534, −6.53348588237262236850057329325, −4.90659799220475828249320606046, 0.54865171335517690371365106374, 2.99806685421999627384279384757, 5.42037893774710437139966590847, 8.735681886020647477304056589957, 10.41728212494074611936827364797, 11.62713701540404592803096463660, 12.55232759386570835442751470637, 14.17027591392097058209219092996, 16.41462218292866141089576262264, 17.86776285277926884065189987196

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