Properties

Degree 2
Conductor 13
Sign $-0.348 - 0.937i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.48 − 0.666i)2-s + (−16.9 + 29.2i)3-s + (−49.6 + 28.6i)4-s + (34.0 + 34.0i)5-s + (−22.5 + 84.0i)6-s + (190. + 51.0i)7-s + (−220. + 220. i)8-s + (−207. − 358. i)9-s + (107. + 62.0i)10-s + (−120. − 451. i)11-s − 1.94e3i·12-s + (2.15e3 + 422. i)13-s + 507.·14-s + (−1.57e3 + 421. i)15-s + (1.43e3 − 2.48e3i)16-s + (−1.34e3 + 775. i)17-s + ⋯
L(s)  = 1  + (0.310 − 0.0832i)2-s + (−0.626 + 1.08i)3-s + (−0.776 + 0.448i)4-s + (0.272 + 0.272i)5-s + (−0.104 + 0.389i)6-s + (0.554 + 0.148i)7-s + (−0.431 + 0.431i)8-s + (−0.284 − 0.492i)9-s + (0.107 + 0.0620i)10-s + (−0.0908 − 0.339i)11-s − 1.12i·12-s + (0.981 + 0.192i)13-s + 0.184·14-s + (−0.466 + 0.124i)15-s + (0.350 − 0.606i)16-s + (−0.273 + 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $-0.348 - 0.937i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ -0.348 - 0.937i)$
$L(\frac{7}{2})$  $\approx$  $0.642225 + 0.923884i$
$L(\frac12)$  $\approx$  $0.642225 + 0.923884i$
$L(4)$   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 \( 1 + (-2.15e3 - 422. i)T \)
good2 \( 1 + (-2.48 + 0.666i)T + (55.4 - 32i)T^{2} \)
3 \( 1 + (16.9 - 29.2i)T + (-364.5 - 631. i)T^{2} \)
5 \( 1 + (-34.0 - 34.0i)T + 1.56e4iT^{2} \)
7 \( 1 + (-190. - 51.0i)T + (1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (120. + 451. i)T + (-1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (1.34e3 - 775. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (3.12e3 - 1.16e4i)T + (-4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (-1.78e4 - 1.02e4i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.11e4 + 1.93e4i)T + (-2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-1.68e3 - 1.68e3i)T + 8.87e8iT^{2} \)
37 \( 1 + (7.65e3 + 2.85e4i)T + (-2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (1.96e4 - 5.27e3i)T + (4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (6.01e4 - 3.46e4i)T + (3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.33e5 + 1.33e5i)T - 1.07e10iT^{2} \)
53 \( 1 - 1.06e5T + 2.21e10T^{2} \)
59 \( 1 + (-3.65e5 - 9.79e4i)T + (3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (-1.00e5 - 1.74e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.25e5 - 6.03e4i)T + (7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (8.49e4 - 3.17e5i)T + (-1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (4.07e4 - 4.07e4i)T - 1.51e11iT^{2} \)
79 \( 1 - 4.35e5T + 2.43e11T^{2} \)
83 \( 1 + (2.56e5 + 2.56e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (2.42e5 + 9.03e5i)T + (-4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (9.07e4 - 3.38e5i)T + (-7.21e11 - 4.16e11i)T^{2} \)
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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

−18.69231864842005385349217232841, −17.47889022854587613790025687523, −16.35533393633986280317523537569, −14.83087960546457096967308066341, −13.43502983555844956534788527635, −11.62696715221710043742483490392, −10.24688499554984614295870381130, −8.550710556007713442668153807593, −5.56296686222079682917202899253, −3.98366223345054869971854439584, 0.971244595941493625544782531415, 5.02115843258048243022037824192, 6.71577173816239441050857580547, 8.875837172356587746691363232238, 11.02025952601841300156116536758, 12.78854876217651741096696642178, 13.54265992861260827454017914748, 15.17179419389670043903665263690, 17.30097503745853293559000413464, 18.05313121819834256257068530803

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