Properties

Degree 2
Conductor 5
Sign $1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 83.7·2-s − 503.·3-s + 4.96e3·4-s − 3.12e3·5-s + 4.21e4·6-s + 1.59e4·7-s − 2.44e5·8-s + 7.60e4·9-s + 2.61e5·10-s + 3.39e5·11-s − 2.49e6·12-s + 2.02e6·13-s − 1.33e6·14-s + 1.57e6·15-s + 1.02e7·16-s − 2.45e6·17-s − 6.37e6·18-s − 4.08e6·19-s − 1.55e7·20-s − 8.03e6·21-s − 2.84e7·22-s + 2.86e7·23-s + 1.22e8·24-s + 9.76e6·25-s − 1.69e8·26-s + 5.08e7·27-s + 7.92e7·28-s + ⋯
L(s)  = 1  − 1.85·2-s − 1.19·3-s + 2.42·4-s − 0.447·5-s + 2.21·6-s + 0.359·7-s − 2.63·8-s + 0.429·9-s + 0.827·10-s + 0.636·11-s − 2.89·12-s + 1.51·13-s − 0.664·14-s + 0.534·15-s + 2.44·16-s − 0.418·17-s − 0.794·18-s − 0.378·19-s − 1.08·20-s − 0.429·21-s − 1.17·22-s + 0.928·23-s + 3.14·24-s + 0.199·25-s − 2.79·26-s + 0.682·27-s + 0.870·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $0.401542$
$L(\frac12)$  $\approx$  $0.401542$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + 3.12e3T \)
good2 \( 1 + 83.7T + 2.04e3T^{2} \)
3 \( 1 + 503.T + 1.77e5T^{2} \)
7 \( 1 - 1.59e4T + 1.97e9T^{2} \)
11 \( 1 - 3.39e5T + 2.85e11T^{2} \)
13 \( 1 - 2.02e6T + 1.79e12T^{2} \)
17 \( 1 + 2.45e6T + 3.42e13T^{2} \)
19 \( 1 + 4.08e6T + 1.16e14T^{2} \)
23 \( 1 - 2.86e7T + 9.52e14T^{2} \)
29 \( 1 + 9.41e6T + 1.22e16T^{2} \)
31 \( 1 - 2.99e8T + 2.54e16T^{2} \)
37 \( 1 + 4.57e8T + 1.77e17T^{2} \)
41 \( 1 - 1.83e8T + 5.50e17T^{2} \)
43 \( 1 - 6.56e8T + 9.29e17T^{2} \)
47 \( 1 + 1.97e8T + 2.47e18T^{2} \)
53 \( 1 - 5.15e9T + 9.26e18T^{2} \)
59 \( 1 + 6.62e8T + 3.01e19T^{2} \)
61 \( 1 - 5.58e8T + 4.35e19T^{2} \)
67 \( 1 - 1.01e10T + 1.22e20T^{2} \)
71 \( 1 - 1.78e10T + 2.31e20T^{2} \)
73 \( 1 + 2.33e10T + 3.13e20T^{2} \)
79 \( 1 - 1.24e10T + 7.47e20T^{2} \)
83 \( 1 - 3.37e10T + 1.28e21T^{2} \)
89 \( 1 - 2.94e10T + 2.77e21T^{2} \)
97 \( 1 + 1.13e11T + 7.15e21T^{2} \)
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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

−20.86775464936115112518076261467, −19.25371857873387388844211070691, −17.93453666412016046820795198294, −16.95574769156702062941203620297, −15.70964955423081921401802810039, −11.68400555027305447688553193978, −10.75608707608599422681900499338, −8.599382500358420294892083163862, −6.52847333889158114408228719657, −0.914486374519612652319967091132, 0.914486374519612652319967091132, 6.52847333889158114408228719657, 8.599382500358420294892083163862, 10.75608707608599422681900499338, 11.68400555027305447688553193978, 15.70964955423081921401802810039, 16.95574769156702062941203620297, 17.93453666412016046820795198294, 19.25371857873387388844211070691, 20.86775464936115112518076261467

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