Properties

Degree 2
Conductor 13
Sign $0.751 - 0.659i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.397 + 1.48i)2-s + (−0.503 − 0.872i)3-s + (53.3 + 30.8i)4-s + (84.5 + 84.5i)5-s + (1.49 − 0.400i)6-s + (41.0 + 153. i)7-s + (−136. + 136. i)8-s + (363. − 630. i)9-s + (−159. + 91.8i)10-s + (−1.01e3 − 270. i)11-s − 62.0i·12-s + (−1.38e3 − 1.70e3i)13-s − 243.·14-s + (31.1 − 116. i)15-s + (1.82e3 + 3.15e3i)16-s + (−1.49e3 − 863. i)17-s + ⋯
L(s)  = 1  + (−0.0497 + 0.185i)2-s + (−0.0186 − 0.0323i)3-s + (0.834 + 0.481i)4-s + (0.676 + 0.676i)5-s + (0.00692 − 0.00185i)6-s + (0.119 + 0.446i)7-s + (−0.266 + 0.266i)8-s + (0.499 − 0.864i)9-s + (−0.159 + 0.0918i)10-s + (−0.759 − 0.203i)11-s − 0.0359i·12-s + (−0.629 − 0.776i)13-s − 0.0887·14-s + (0.00923 − 0.0344i)15-s + (0.445 + 0.771i)16-s + (−0.304 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.751 - 0.659i$
motivic weight  =  \(6\)
character  :  $\chi_{13} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 13,\ (\ :3),\ 0.751 - 0.659i)$
$L(\frac{7}{2})$  $\approx$  $1.52907 + 0.575408i$
$L(\frac12)$  $\approx$  $1.52907 + 0.575408i$
$L(4)$   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 \( 1 + (1.38e3 + 1.70e3i)T \)
good2 \( 1 + (0.397 - 1.48i)T + (-55.4 - 32i)T^{2} \)
3 \( 1 + (0.503 + 0.872i)T + (-364.5 + 631. i)T^{2} \)
5 \( 1 + (-84.5 - 84.5i)T + 1.56e4iT^{2} \)
7 \( 1 + (-41.0 - 153. i)T + (-1.01e5 + 5.88e4i)T^{2} \)
11 \( 1 + (1.01e3 + 270. i)T + (1.53e6 + 8.85e5i)T^{2} \)
17 \( 1 + (1.49e3 + 863. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-6.57e3 + 1.76e3i)T + (4.07e7 - 2.35e7i)T^{2} \)
23 \( 1 + (3.37e3 - 1.94e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (1.75e4 + 3.04e4i)T + (-2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.21e4 + 1.21e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (-5.55e4 - 1.48e4i)T + (2.22e9 + 1.28e9i)T^{2} \)
41 \( 1 + (1.95e4 - 7.29e4i)T + (-4.11e9 - 2.37e9i)T^{2} \)
43 \( 1 + (-5.62e4 - 3.24e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (6.42e4 - 6.42e4i)T - 1.07e10iT^{2} \)
53 \( 1 + 2.72e5T + 2.21e10T^{2} \)
59 \( 1 + (-7.41e4 - 2.76e5i)T + (-3.65e10 + 2.10e10i)T^{2} \)
61 \( 1 + (-9.89e4 + 1.71e5i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.49e5 - 5.57e5i)T + (-7.83e10 - 4.52e10i)T^{2} \)
71 \( 1 + (-3.66e5 + 9.83e4i)T + (1.10e11 - 6.40e10i)T^{2} \)
73 \( 1 + (-1.88e5 + 1.88e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 7.16e5T + 2.43e11T^{2} \)
83 \( 1 + (4.06e5 + 4.06e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (-2.70e5 - 7.25e4i)T + (4.30e11 + 2.48e11i)T^{2} \)
97 \( 1 + (2.54e5 - 6.82e4i)T + (7.21e11 - 4.16e11i)T^{2} \)
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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

−18.39204833032226145923574406681, −17.53274782416388982729608104415, −15.84279829094426127000871617166, −14.84637355319914691820561565881, −12.96593444328320557343836686552, −11.48723329625472273198810087239, −9.847164343735793462609428091780, −7.62815827485183607604198531862, −6.05291243005062027288465206815, −2.71114265878497823920794529964, 1.82174535677394454174697630032, 5.22411451123131558421013549334, 7.32491982290296136818012629031, 9.669324310246991226505791062857, 10.91964620325629180831908929509, 12.69791827816180474573431373998, 14.14899458453662414160985869561, 15.88440885226180024761095051324, 16.85495451828104173185104723231, 18.55395527011050969187548729364

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