Properties

Degree 2
Conductor 13
Sign $0.804 - 0.593i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.37 + 2.51i)2-s + (10.4 − 18.0i)3-s + (26.0 − 15.0i)4-s + (160. + 160. i)5-s + (−52.2 + 195. i)6-s + (162. + 43.4i)7-s + (232. − 232. i)8-s + (147. + 255. i)9-s + (−1.90e3 − 1.09e3i)10-s + (−264. − 986. i)11-s − 627. i·12-s + (189. + 2.18e3i)13-s − 1.62e3·14-s + (4.55e3 − 1.21e3i)15-s + (−2.55e3 + 4.43e3i)16-s + (1.83e3 − 1.06e3i)17-s + ⋯
L(s)  = 1  + (−1.17 + 0.313i)2-s + (0.385 − 0.667i)3-s + (0.407 − 0.235i)4-s + (1.28 + 1.28i)5-s + (−0.242 + 0.903i)6-s + (0.472 + 0.126i)7-s + (0.453 − 0.453i)8-s + (0.202 + 0.351i)9-s + (−1.90 − 1.09i)10-s + (−0.198 − 0.741i)11-s − 0.363i·12-s + (0.0861 + 0.996i)13-s − 0.593·14-s + (1.34 − 0.361i)15-s + (−0.624 + 1.08i)16-s + (0.373 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.804 - 0.593i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ 0.804 - 0.593i)$
$L(\frac{7}{2})$  $\approx$  $0.986169 + 0.324440i$
$L(\frac12)$  $\approx$  $0.986169 + 0.324440i$
$L(4)$   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 \( 1 + (-189. - 2.18e3i)T \)
good2 \( 1 + (9.37 - 2.51i)T + (55.4 - 32i)T^{2} \)
3 \( 1 + (-10.4 + 18.0i)T + (-364.5 - 631. i)T^{2} \)
5 \( 1 + (-160. - 160. i)T + 1.56e4iT^{2} \)
7 \( 1 + (-162. - 43.4i)T + (1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (264. + 986. i)T + (-1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (-1.83e3 + 1.06e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-334. + 1.24e3i)T + (-4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (1.19e4 + 6.88e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.20e4 + 2.09e4i)T + (-2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (2.63e4 + 2.63e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (7.41e3 + 2.76e4i)T + (-2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (-8.34e4 + 2.23e4i)T + (4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (1.02e5 - 5.93e4i)T + (3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-2.43e3 + 2.43e3i)T - 1.07e10iT^{2} \)
53 \( 1 + 2.62e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.48e5 - 3.97e4i)T + (3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (-1.47e4 - 2.55e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-8.60e4 + 2.30e4i)T + (7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (-1.72e5 + 6.44e5i)T + (-1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (-4.65e4 + 4.65e4i)T - 1.51e11iT^{2} \)
79 \( 1 - 1.69e5T + 2.43e11T^{2} \)
83 \( 1 + (-3.78e5 - 3.78e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (-3.73e4 - 1.39e5i)T + (-4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (5.46e4 - 2.04e5i)T + (-7.21e11 - 4.16e11i)T^{2} \)
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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

−18.50925484609581690514847074043, −17.81992077725998998067122395207, −16.41157249827104671709150136106, −14.32581329701405988783902283893, −13.46007418946369790090683879820, −10.88335456092995078735079627378, −9.570298169040511965528498926014, −7.898324083103360429765335457832, −6.49776928575229438495238041488, −2.00385189246237898609294475690, 1.43336555171954563279725081331, 5.06656412919901956394396365986, 8.310804590919634204518780416288, 9.515567957571038482823215754995, 10.26746839551250006577749401701, 12.65429733155437300085186038401, 14.27522414108012846007348362405, 16.10681305692999957948033782820, 17.46587456316887448218797118072, 18.01494907634907124109982867934

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