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  − 1.26e5·2-s − 8.63e7·3-s + 7.38e9·4-s + 1.52e11·5-s + 1.09e13·6-s + 7.78e13·7-s + 1.52e14·8-s + 1.89e15·9-s − 1.92e16·10-s + 1.61e17·11-s − 6.37e17·12-s − 2.28e18·13-s − 9.84e18·14-s − 1.31e19·15-s − 8.26e19·16-s + 2.46e20·17-s − 2.39e20·18-s − 1.53e21·19-s + 1.12e21·20-s − 6.72e21·21-s − 2.04e22·22-s − 2.16e22·23-s − 1.31e22·24-s + 2.32e22·25-s + 2.89e23·26-s + 3.16e23·27-s + 5.75e23·28-s + ⋯
L(s)  = 1  − 1.36·2-s − 1.15·3-s + 0.859·4-s + 0.447·5-s + 1.57·6-s + 0.885·7-s + 0.190·8-s + 0.340·9-s − 0.609·10-s + 1.06·11-s − 0.995·12-s − 0.953·13-s − 1.20·14-s − 0.517·15-s − 1.12·16-s + 1.22·17-s − 0.464·18-s − 1.22·19-s + 0.384·20-s − 1.02·21-s − 1.44·22-s − 0.735·23-s − 0.221·24-s + 0.200·25-s + 1.30·26-s + 0.763·27-s + 0.761·28-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  =  $(2,\ 5,\ (\ :33/2),\ 1)$
$L(17)$  $\approx$  $0.6489982189$
$L(\frac12)$  $\approx$  $0.6489982189$
$L(\frac{35}{2})$   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 \( 1 + 1.26e5T + 8.58e9T^{2} \)
3 \( 1 + 8.63e7T + 5.55e15T^{2} \)
7 \( 1 - 7.78e13T + 7.73e27T^{2} \)
11 \( 1 - 1.61e17T + 2.32e34T^{2} \)
13 \( 1 + 2.28e18T + 5.75e36T^{2} \)
17 \( 1 - 2.46e20T + 4.02e40T^{2} \)
19 \( 1 + 1.53e21T + 1.58e42T^{2} \)
23 \( 1 + 2.16e22T + 8.65e44T^{2} \)
29 \( 1 + 2.07e23T + 1.81e48T^{2} \)
31 \( 1 - 1.75e24T + 1.64e49T^{2} \)
37 \( 1 - 8.12e25T + 5.63e51T^{2} \)
41 \( 1 + 1.92e26T + 1.66e53T^{2} \)
43 \( 1 - 8.01e26T + 8.02e53T^{2} \)
47 \( 1 + 7.53e27T + 1.51e55T^{2} \)
53 \( 1 - 1.75e28T + 7.96e56T^{2} \)
59 \( 1 + 2.97e29T + 2.74e58T^{2} \)
61 \( 1 + 3.70e29T + 8.23e58T^{2} \)
67 \( 1 - 8.43e29T + 1.82e60T^{2} \)
71 \( 1 - 5.92e30T + 1.23e61T^{2} \)
73 \( 1 - 7.49e30T + 3.08e61T^{2} \)
79 \( 1 - 6.07e30T + 4.18e62T^{2} \)
83 \( 1 - 3.47e31T + 2.13e63T^{2} \)
89 \( 1 - 1.80e32T + 2.13e64T^{2} \)
97 \( 1 - 1.09e33T + 3.65e65T^{2} \)
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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

−16.76199098162569760365475918922, −14.44994961692951512737681442002, −12.05305686940475542548586740387, −10.85245258198327515084305817331, −9.607200677611740859295500562729, −8.002717772307227546656617874337, −6.36637790558535352432706740019, −4.75978964464573230976129542578, −1.81880437593070529066471579048, −0.66056961295763754977032736231, 0.66056961295763754977032736231, 1.81880437593070529066471579048, 4.75978964464573230976129542578, 6.36637790558535352432706740019, 8.002717772307227546656617874337, 9.607200677611740859295500562729, 10.85245258198327515084305817331, 12.05305686940475542548586740387, 14.44994961692951512737681442002, 16.76199098162569760365475918922

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