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Properties

Label 2-13-13.12-c9-0-3
Degree $2$
Conductor $13$
Sign $-0.995 - 0.0949i$
Analytic cond. $6.69546$
Root an. cond. $2.58755$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 36.6i·2-s + 126.·3-s − 829.·4-s + 2.09e3i·5-s + 4.65e3i·6-s − 3.35e3i·7-s − 1.16e4i·8-s − 3.55e3·9-s − 7.67e4·10-s − 3.02e4i·11-s − 1.05e5·12-s + (1.02e5 + 9.78e3i)13-s + 1.22e5·14-s + 2.66e5i·15-s + 847.·16-s + 1.66e4·17-s + ⋯
L(s)  = 1  + 1.61i·2-s + 0.905·3-s − 1.61·4-s + 1.49i·5-s + 1.46i·6-s − 0.527i·7-s − 1.00i·8-s − 0.180·9-s − 2.42·10-s − 0.621i·11-s − 1.46·12-s + (0.995 + 0.0949i)13-s + 0.854·14-s + 1.35i·15-s + 0.00323·16-s + 0.0484·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0949i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.995 - 0.0949i$
Analytic conductor: \(6.69546\)
Root analytic conductor: \(2.58755\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :9/2),\ -0.995 - 0.0949i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0890652 + 1.87087i\)
\(L(\frac12)\) \(\approx\) \(0.0890652 + 1.87087i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.02e5 - 9.78e3i)T \)
good2 \( 1 - 36.6iT - 512T^{2} \)
3 \( 1 - 126.T + 1.96e4T^{2} \)
5 \( 1 - 2.09e3iT - 1.95e6T^{2} \)
7 \( 1 + 3.35e3iT - 4.03e7T^{2} \)
11 \( 1 + 3.02e4iT - 2.35e9T^{2} \)
17 \( 1 - 1.66e4T + 1.18e11T^{2} \)
19 \( 1 - 9.72e5iT - 3.22e11T^{2} \)
23 \( 1 - 2.50e6T + 1.80e12T^{2} \)
29 \( 1 - 4.39e6T + 1.45e13T^{2} \)
31 \( 1 - 3.88e6iT - 2.64e13T^{2} \)
37 \( 1 + 8.44e6iT - 1.29e14T^{2} \)
41 \( 1 + 9.82e6iT - 3.27e14T^{2} \)
43 \( 1 - 1.24e7T + 5.02e14T^{2} \)
47 \( 1 + 2.73e7iT - 1.11e15T^{2} \)
53 \( 1 + 4.22e7T + 3.29e15T^{2} \)
59 \( 1 - 2.26e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.70e8T + 1.16e16T^{2} \)
67 \( 1 + 4.12e5iT - 2.72e16T^{2} \)
71 \( 1 + 2.40e8iT - 4.58e16T^{2} \)
73 \( 1 + 3.37e8iT - 5.88e16T^{2} \)
79 \( 1 + 1.13e8T + 1.19e17T^{2} \)
83 \( 1 - 5.09e8iT - 1.86e17T^{2} \)
89 \( 1 + 4.18e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.22e9iT - 7.60e17T^{2} \)
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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

−18.20981248412863687796239164049, −16.75220495744961901038050847954, −15.36239809862658829406034735837, −14.32831984120264013039052101723, −13.80322834457947577244573513850, −10.76808135582743352297690246005, −8.711348364633175660673856509886, −7.43870275949815888705418467051, −6.13466625895213349378168069059, −3.37158820459940897695569841675, 1.05428497206120721649969871003, 2.77312506341140626402271146247, 4.68399068777890171775510588567, 8.668856699017719384478027268124, 9.303192833167714712302811850852, 11.33543253218959589326696697561, 12.71880655198529044444180864860, 13.50897546040615358085908378375, 15.49382792723207697686115102905, 17.41689722508960859283882390018

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