Properties

Degree 2
Conductor 5
Sign $1$
Motivic weight 33
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.78e5·2-s + 8.55e7·3-s + 2.34e10·4-s + 1.52e11·5-s + 1.52e13·6-s − 8.09e13·7-s + 2.64e15·8-s + 1.75e15·9-s + 2.72e16·10-s + 1.95e17·11-s + 2.00e18·12-s − 2.59e18·13-s − 1.44e19·14-s + 1.30e19·15-s + 2.72e20·16-s − 1.52e20·17-s + 3.13e20·18-s + 1.39e21·19-s + 3.57e21·20-s − 6.92e21·21-s + 3.50e22·22-s − 3.40e22·23-s + 2.26e23·24-s + 2.32e22·25-s − 4.63e23·26-s − 3.25e23·27-s − 1.89e24·28-s + ⋯
L(s)  = 1  + 1.92·2-s + 1.14·3-s + 2.72·4-s + 0.447·5-s + 2.21·6-s − 0.920·7-s + 3.32·8-s + 0.315·9-s + 0.863·10-s + 1.28·11-s + 3.12·12-s − 1.08·13-s − 1.77·14-s + 0.512·15-s + 3.69·16-s − 0.758·17-s + 0.608·18-s + 1.10·19-s + 1.21·20-s − 1.05·21-s + 2.47·22-s − 1.15·23-s + 3.81·24-s + 0.200·25-s − 2.08·26-s − 0.785·27-s − 2.50·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(33\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :33/2),\ 1)$
$L(17)$  $\approx$  $9.841317161$
$L(\frac12)$  $\approx$  $9.841317161$
$L(\frac{35}{2})$   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 \( 1 - 1.52e11T \)
good2 \( 1 - 1.78e5T + 8.58e9T^{2} \)
3 \( 1 - 8.55e7T + 5.55e15T^{2} \)
7 \( 1 + 8.09e13T + 7.73e27T^{2} \)
11 \( 1 - 1.95e17T + 2.32e34T^{2} \)
13 \( 1 + 2.59e18T + 5.75e36T^{2} \)
17 \( 1 + 1.52e20T + 4.02e40T^{2} \)
19 \( 1 - 1.39e21T + 1.58e42T^{2} \)
23 \( 1 + 3.40e22T + 8.65e44T^{2} \)
29 \( 1 + 1.49e24T + 1.81e48T^{2} \)
31 \( 1 + 1.38e24T + 1.64e49T^{2} \)
37 \( 1 - 1.15e26T + 5.63e51T^{2} \)
41 \( 1 + 6.35e25T + 1.66e53T^{2} \)
43 \( 1 + 2.64e26T + 8.02e53T^{2} \)
47 \( 1 - 1.03e27T + 1.51e55T^{2} \)
53 \( 1 - 5.20e28T + 7.96e56T^{2} \)
59 \( 1 + 6.17e28T + 2.74e58T^{2} \)
61 \( 1 + 1.21e29T + 8.23e58T^{2} \)
67 \( 1 + 2.87e29T + 1.82e60T^{2} \)
71 \( 1 + 2.84e30T + 1.23e61T^{2} \)
73 \( 1 + 2.67e30T + 3.08e61T^{2} \)
79 \( 1 + 7.84e30T + 4.18e62T^{2} \)
83 \( 1 - 1.28e31T + 2.13e63T^{2} \)
89 \( 1 + 1.56e32T + 2.13e64T^{2} \)
97 \( 1 - 4.84e32T + 3.65e65T^{2} \)
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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

−15.02176375737561015324114368262, −14.13392198410056059317031652627, −13.17214185603765693534541201889, −11.79243316994726270369526069385, −9.588808615860684789290250184138, −7.22296820316263386183858652121, −5.89530756446408114783769280067, −4.10720691939794766588919921973, −3.02921678763293403068550536557, −1.97942743588026212141340366008, 1.97942743588026212141340366008, 3.02921678763293403068550536557, 4.10720691939794766588919921973, 5.89530756446408114783769280067, 7.22296820316263386183858652121, 9.588808615860684789290250184138, 11.79243316994726270369526069385, 13.17214185603765693534541201889, 14.13392198410056059317031652627, 15.02176375737561015324114368262

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