Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29e5·2-s − 6.82e7·3-s + 8.07e9·4-s − 1.52e11·5-s + 8.81e12·6-s − 1.39e14·7-s + 6.68e13·8-s − 8.95e14·9-s + 1.96e16·10-s − 1.11e17·11-s − 5.51e17·12-s + 3.82e18·13-s + 1.79e19·14-s + 1.04e19·15-s − 7.79e19·16-s + 1.35e20·17-s + 1.15e20·18-s + 1.50e21·19-s − 1.23e21·20-s + 9.51e21·21-s + 1.44e22·22-s + 2.26e22·23-s − 4.56e21·24-s + 2.32e22·25-s − 4.93e23·26-s + 4.40e23·27-s − 1.12e24·28-s + ⋯
L(s)  = 1  − 1.39·2-s − 0.915·3-s + 0.939·4-s − 0.447·5-s + 1.27·6-s − 1.58·7-s + 0.0839·8-s − 0.161·9-s + 0.622·10-s − 0.733·11-s − 0.860·12-s + 1.59·13-s + 2.20·14-s + 0.409·15-s − 1.05·16-s + 0.676·17-s + 0.224·18-s + 1.20·19-s − 0.420·20-s + 1.45·21-s + 1.02·22-s + 0.769·23-s − 0.0769·24-s + 0.200·25-s − 2.22·26-s + 1.06·27-s − 1.48·28-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  =  1
Selberg data  =  $(2,\ 5,\ (\ :33/2),\ -1)$
$L(17)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{35}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + 1.52e11T \)
good2 \( 1 + 1.29e5T + 8.58e9T^{2} \)
3 \( 1 + 6.82e7T + 5.55e15T^{2} \)
7 \( 1 + 1.39e14T + 7.73e27T^{2} \)
11 \( 1 + 1.11e17T + 2.32e34T^{2} \)
13 \( 1 - 3.82e18T + 5.75e36T^{2} \)
17 \( 1 - 1.35e20T + 4.02e40T^{2} \)
19 \( 1 - 1.50e21T + 1.58e42T^{2} \)
23 \( 1 - 2.26e22T + 8.65e44T^{2} \)
29 \( 1 + 6.82e23T + 1.81e48T^{2} \)
31 \( 1 + 5.41e24T + 1.64e49T^{2} \)
37 \( 1 + 7.55e24T + 5.63e51T^{2} \)
41 \( 1 + 1.95e26T + 1.66e53T^{2} \)
43 \( 1 - 5.44e26T + 8.02e53T^{2} \)
47 \( 1 + 3.58e27T + 1.51e55T^{2} \)
53 \( 1 - 4.28e28T + 7.96e56T^{2} \)
59 \( 1 + 2.15e28T + 2.74e58T^{2} \)
61 \( 1 - 5.24e29T + 8.23e58T^{2} \)
67 \( 1 + 1.47e30T + 1.82e60T^{2} \)
71 \( 1 + 1.24e30T + 1.23e61T^{2} \)
73 \( 1 + 9.14e30T + 3.08e61T^{2} \)
79 \( 1 + 2.52e31T + 4.18e62T^{2} \)
83 \( 1 + 1.28e31T + 2.13e63T^{2} \)
89 \( 1 - 1.22e32T + 2.13e64T^{2} \)
97 \( 1 - 2.05e31T + 3.65e65T^{2} \)
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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

−16.01417531583702851255545789527, −13.13848939401902070662832303057, −11.38197413170177910578789601639, −10.22107690129088085391143206879, −8.828679055593915378418729351198, −7.18164491004867355002602034511, −5.72581932017110231922042543767, −3.29324260045397388126301025346, −0.945238004110062940083758615377, 0, 0.945238004110062940083758615377, 3.29324260045397388126301025346, 5.72581932017110231922042543767, 7.18164491004867355002602034511, 8.828679055593915378418729351198, 10.22107690129088085391143206879, 11.38197413170177910578789601639, 13.13848939401902070662832303057, 16.01417531583702851255545789527

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