Properties

Degree 2
Conductor 7
Sign $0.120 + 0.992i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.54 − 6.13i)2-s + (5.04 − 8.73i)3-s + (−9.08 + 15.7i)4-s + (39.9 + 69.1i)5-s − 71.4·6-s + (43.1 − 122. i)7-s − 97.9·8-s + (70.6 + 122. i)9-s + (282. − 489. i)10-s + (−175. + 304. i)11-s + (91.5 + 158. i)12-s − 291.·13-s + (−902. + 168. i)14-s + 804.·15-s + (637. + 1.10e3i)16-s + (185. − 320. i)17-s + ⋯
L(s)  = 1  + (−0.626 − 1.08i)2-s + (0.323 − 0.560i)3-s + (−0.283 + 0.491i)4-s + (0.713 + 1.23i)5-s − 0.809·6-s + (0.332 − 0.942i)7-s − 0.541·8-s + (0.290 + 0.503i)9-s + (0.893 − 1.54i)10-s + (−0.438 + 0.759i)11-s + (0.183 + 0.317i)12-s − 0.478·13-s + (−1.23 + 0.229i)14-s + 0.923·15-s + (0.622 + 1.07i)16-s + (0.155 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7\)
\( \varepsilon \)  =  $0.120 + 0.992i$
motivic weight  =  \(5\)
character  :  $\chi_{7} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 7,\ (\ :5/2),\ 0.120 + 0.992i)$
$L(3)$  $\approx$  $0.697334 - 0.617778i$
$L(\frac12)$  $\approx$  $0.697334 - 0.617778i$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 7$, \(F_p\) is a polynomial of degree 2. If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + (-43.1 + 122. i)T \)
good2 \( 1 + (3.54 + 6.13i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-5.04 + 8.73i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-39.9 - 69.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (175. - 304. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 291.T + 3.71e5T^{2} \)
17 \( 1 + (-185. + 320. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (752. + 1.30e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-212. - 368. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 7.78e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.28e3 + 2.23e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (369. + 640. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 7.02e3T + 1.15e8T^{2} \)
43 \( 1 - 1.83e3T + 1.47e8T^{2} \)
47 \( 1 + (-766. - 1.32e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-4.76e3 + 8.25e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.48e4 + 2.56e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.32e4 - 4.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.33e4 - 2.31e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.50e4 + 6.07e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.35e4 - 2.34e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.97e4T + 3.93e9T^{2} \)
89 \( 1 + (2.17e4 + 3.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.03e5T + 8.58e9T^{2} \)
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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

−20.92913808019726655441849095092, −19.61857227158611550287685353359, −18.52623624063460036136455287619, −17.52976270108965941990062847150, −14.70834586642623482984680703603, −13.17891789008517935944615106046, −10.99748335226301824877763326540, −9.946388780896094143993725974526, −7.25821217549352394679105745722, −2.24580681910652224236344448604, 5.64713656488976520025843190553, 8.435416633558901220546111564259, 9.454139797789076397700060710186, 12.52412358353080258310017563007, 14.77321029738825350425838697111, 16.03801878611397430936717703502, 17.12985055423116886193794034122, 18.52592151294677291054043976735, 20.82031395049526902366094219680, 21.48465210657445302005626883633

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