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  :  $\chi_{13} (2, \cdot )$
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

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

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