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  + 7.64e4·2-s + 7.43e7·3-s − 2.75e9·4-s + 1.52e11·5-s + 5.67e12·6-s + 1.36e14·7-s − 8.66e14·8-s − 3.57e13·9-s + 1.16e16·10-s + 3.84e16·11-s − 2.04e17·12-s + 2.57e18·13-s + 1.04e19·14-s + 1.13e19·15-s − 4.25e19·16-s + 1.44e20·17-s − 2.73e18·18-s − 2.79e20·19-s − 4.19e20·20-s + 1.01e22·21-s + 2.93e21·22-s + 3.71e22·23-s − 6.44e22·24-s + 2.32e22·25-s + 1.96e23·26-s − 4.15e23·27-s − 3.76e23·28-s + ⋯
L(s)  = 1  + 0.824·2-s + 0.996·3-s − 0.320·4-s + 0.447·5-s + 0.821·6-s + 1.55·7-s − 1.08·8-s − 0.00642·9-s + 0.368·10-s + 0.252·11-s − 0.319·12-s + 1.07·13-s + 1.28·14-s + 0.445·15-s − 0.577·16-s + 0.720·17-s − 0.00530·18-s − 0.222·19-s − 0.143·20-s + 1.55·21-s + 0.207·22-s + 1.26·23-s − 1.08·24-s + 0.200·25-s + 0.883·26-s − 1.00·27-s − 0.499·28-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  =  $(2,\ 5,\ (\ :33/2),\ 1)$
$L(17)$  $\approx$  $4.846485149$
$L(\frac12)$  $\approx$  $4.846485149$
$L(\frac{35}{2})$   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 \( 1 - 7.64e4T + 8.58e9T^{2} \)
3 \( 1 - 7.43e7T + 5.55e15T^{2} \)
7 \( 1 - 1.36e14T + 7.73e27T^{2} \)
11 \( 1 - 3.84e16T + 2.32e34T^{2} \)
13 \( 1 - 2.57e18T + 5.75e36T^{2} \)
17 \( 1 - 1.44e20T + 4.02e40T^{2} \)
19 \( 1 + 2.79e20T + 1.58e42T^{2} \)
23 \( 1 - 3.71e22T + 8.65e44T^{2} \)
29 \( 1 + 1.90e24T + 1.81e48T^{2} \)
31 \( 1 - 5.88e24T + 1.64e49T^{2} \)
37 \( 1 - 1.05e26T + 5.63e51T^{2} \)
41 \( 1 + 2.64e25T + 1.66e53T^{2} \)
43 \( 1 - 1.20e27T + 8.02e53T^{2} \)
47 \( 1 + 3.53e27T + 1.51e55T^{2} \)
53 \( 1 + 3.25e28T + 7.96e56T^{2} \)
59 \( 1 - 3.41e28T + 2.74e58T^{2} \)
61 \( 1 + 6.01e28T + 8.23e58T^{2} \)
67 \( 1 + 2.17e30T + 1.82e60T^{2} \)
71 \( 1 - 9.20e29T + 1.23e61T^{2} \)
73 \( 1 + 4.72e30T + 3.08e61T^{2} \)
79 \( 1 + 2.14e31T + 4.18e62T^{2} \)
83 \( 1 + 6.60e31T + 2.13e63T^{2} \)
89 \( 1 + 6.08e31T + 2.13e64T^{2} \)
97 \( 1 - 6.12e30T + 3.65e65T^{2} \)
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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

−14.94385545081578960179848780109, −14.26068940582343875210377223832, −13.22180237919657671905070970029, −11.34762409678828319493196456924, −9.147655308607214397601451894542, −8.085821336103776534614337739285, −5.71946333803671060531183616557, −4.33550553521135272297111188040, −2.92267600354425143109178969269, −1.33782873674659544208827752433, 1.33782873674659544208827752433, 2.92267600354425143109178969269, 4.33550553521135272297111188040, 5.71946333803671060531183616557, 8.085821336103776534614337739285, 9.147655308607214397601451894542, 11.34762409678828319493196456924, 13.22180237919657671905070970029, 14.26068940582343875210377223832, 14.94385545081578960179848780109

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