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Properties

Label 2-13-13.8-c8-0-8
Degree $2$
Conductor $13$
Sign $-0.936 - 0.350i$
Analytic cond. $5.29592$
Root an. cond. $2.30128$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.74 − 9.74i)2-s + 19.1·3-s − 65.9i·4-s + (−198. − 198. i)5-s + (−186. − 186. i)6-s + (−1.52e3 + 1.52e3i)7-s + (−3.13e3 + 3.13e3i)8-s − 6.19e3·9-s + 3.87e3i·10-s + (−2.96e3 + 2.96e3i)11-s − 1.26e3i·12-s + (1.82e3 − 2.85e4i)13-s + 2.96e4·14-s + (−3.80e3 − 3.80e3i)15-s + 4.42e4·16-s − 3.23e4i·17-s + ⋯
L(s)  = 1  + (−0.609 − 0.609i)2-s + 0.236·3-s − 0.257i·4-s + (−0.317 − 0.317i)5-s + (−0.144 − 0.144i)6-s + (−0.633 + 0.633i)7-s + (−0.766 + 0.766i)8-s − 0.943·9-s + 0.387i·10-s + (−0.202 + 0.202i)11-s − 0.0609i·12-s + (0.0639 − 0.997i)13-s + 0.771·14-s + (−0.0752 − 0.0752i)15-s + 0.675·16-s − 0.387i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.936 - 0.350i$
Analytic conductor: \(5.29592\)
Root analytic conductor: \(2.30128\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :4),\ -0.936 - 0.350i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0597760 + 0.330371i\)
\(L(\frac12)\) \(\approx\) \(0.0597760 + 0.330371i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.82e3 + 2.85e4i)T \)
good2 \( 1 + (9.74 + 9.74i)T + 256iT^{2} \)
3 \( 1 - 19.1T + 6.56e3T^{2} \)
5 \( 1 + (198. + 198. i)T + 3.90e5iT^{2} \)
7 \( 1 + (1.52e3 - 1.52e3i)T - 5.76e6iT^{2} \)
11 \( 1 + (2.96e3 - 2.96e3i)T - 2.14e8iT^{2} \)
17 \( 1 + 3.23e4iT - 6.97e9T^{2} \)
19 \( 1 + (1.06e4 + 1.06e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 6.04e4iT - 7.83e10T^{2} \)
29 \( 1 + 5.49e5T + 5.00e11T^{2} \)
31 \( 1 + (1.94e4 + 1.94e4i)T + 8.52e11iT^{2} \)
37 \( 1 + (-1.93e6 + 1.93e6i)T - 3.51e12iT^{2} \)
41 \( 1 + (-2.53e5 - 2.53e5i)T + 7.98e12iT^{2} \)
43 \( 1 + 5.09e6iT - 1.16e13T^{2} \)
47 \( 1 + (6.02e6 - 6.02e6i)T - 2.38e13iT^{2} \)
53 \( 1 + 1.10e7T + 6.22e13T^{2} \)
59 \( 1 + (-1.69e6 + 1.69e6i)T - 1.46e14iT^{2} \)
61 \( 1 - 1.11e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.60e7 - 1.60e7i)T + 4.06e14iT^{2} \)
71 \( 1 + (-2.04e7 - 2.04e7i)T + 6.45e14iT^{2} \)
73 \( 1 + (2.81e7 - 2.81e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 6.53e6T + 1.51e15T^{2} \)
83 \( 1 + (3.82e7 + 3.82e7i)T + 2.25e15iT^{2} \)
89 \( 1 + (3.40e7 - 3.40e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (1.29e7 + 1.29e7i)T + 7.83e15iT^{2} \)
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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

−17.56973478783240986722770357101, −15.82434185312360963542065348256, −14.52886966062119126508930656193, −12.61586535532198201096295689496, −11.19275344872922231969389648254, −9.619889439964207670323426223506, −8.367638665664222994666406104104, −5.69711941748115046250431482982, −2.69640119852959524079412431704, −0.22917705208011738588258726606, 3.44127282609567687518556127624, 6.54671318845413788087571711813, 7.995488008434359073582993943894, 9.449476591650485570919706247143, 11.42852557726985183503387461697, 13.23147654896850910082931163517, 14.82666609642922103454668471713, 16.32249087699903786336294539961, 17.12644278048050837978351939004, 18.64038167839018263048522600601

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