Properties

Degree 2
Conductor 13
Sign $-0.830 + 0.557i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.63 − 13.5i)2-s + (−6.58 − 11.4i)3-s + (−115. − 66.6i)4-s + (128. + 128. i)5-s + (−178. + 47.8i)6-s + (−92.7 − 346. i)7-s + (−688. + 688. i)8-s + (277. − 481. i)9-s + (2.20e3 − 1.27e3i)10-s + (172. + 46.1i)11-s + 1.75e3i·12-s + (1.15e3 + 1.87e3i)13-s − 5.03e3·14-s + (618. − 2.30e3i)15-s + (2.57e3 + 4.45e3i)16-s + (7.24e3 + 4.18e3i)17-s + ⋯
L(s)  = 1  + (0.454 − 1.69i)2-s + (−0.243 − 0.422i)3-s + (−1.80 − 1.04i)4-s + (1.02 + 1.02i)5-s + (−0.826 + 0.221i)6-s + (−0.270 − 1.00i)7-s + (−1.34 + 1.34i)8-s + (0.381 − 0.660i)9-s + (2.20 − 1.27i)10-s + (0.129 + 0.0347i)11-s + 1.01i·12-s + (0.524 + 0.851i)13-s − 1.83·14-s + (0.183 − 0.684i)15-s + (0.627 + 1.08i)16-s + (1.47 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $-0.830 + 0.557i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ -0.830 + 0.557i)$
$L(\frac{7}{2})$  $\approx$  $0.476592 - 1.56593i$
$L(\frac12)$  $\approx$  $0.476592 - 1.56593i$
$L(4)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 13$,\(F_p(T)\) is a polynomial of degree 2. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (-1.15e3 - 1.87e3i)T \)
good2 \( 1 + (-3.63 + 13.5i)T + (-55.4 - 32i)T^{2} \)
3 \( 1 + (6.58 + 11.4i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (-128. - 128. i)T + 1.56e4iT^{2} \)
7 \( 1 + (92.7 + 346. i)T + (-1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (-172. - 46.1i)T + (1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (-7.24e3 - 4.18e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (5.54e3 - 1.48e3i)T + (4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (3.93e3 - 2.27e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (6.20e3 + 1.07e4i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-3.69e4 - 3.69e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (2.47e4 + 6.62e3i)T + (2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (-1.47e3 + 5.49e3i)T + (-4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (5.91e4 + 3.41e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (3.45e4 - 3.45e4i)T - 1.07e10iT^{2} \)
53 \( 1 + 1.35e5T + 2.21e10T^{2} \)
59 \( 1 + (4.09e4 + 1.52e5i)T + (-3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (1.12e5 - 1.95e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-7.43e3 + 2.77e4i)T + (-7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (-4.24e5 + 1.13e5i)T + (1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (-8.86e4 + 8.86e4i)T - 1.51e11iT^{2} \)
79 \( 1 - 2.62e5T + 2.43e11T^{2} \)
83 \( 1 + (2.32e5 + 2.32e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (-1.90e5 - 5.10e4i)T + (4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (-2.18e4 + 5.84e3i)T + (7.21e11 - 4.16e11i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.50246777393189396897332120145, −17.27368648687672895229136017736, −14.39092371115483933727092513516, −13.51242302804466100634744738319, −12.19942156348466199026791566850, −10.64029608020504491941128447998, −9.825766608512008368257597754069, −6.43906644078476988296220687880, −3.64732805541704382198508374920, −1.50671643796533895721957085885, 5.02396513143927168776925173483, 5.92892451958388447060706658586, 8.227743932967305371855732812611, 9.640607620922817712511818132293, 12.67256238253201563054053177178, 13.72202142231038127379237953016, 15.31384709635712487340482606654, 16.32811681465441372687696587025, 17.12999364701172020361169488776, 18.50389800267957774968165381918

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