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  + 1.32e5·2-s − 1.17e8·3-s + 9.07e9·4-s + 1.52e11·5-s − 1.56e13·6-s − 1.34e13·7-s + 6.49e13·8-s + 8.25e15·9-s + 2.02e16·10-s − 7.96e16·11-s − 1.06e18·12-s − 1.16e18·13-s − 1.79e18·14-s − 1.79e19·15-s − 6.93e19·16-s + 3.27e20·17-s + 1.09e21·18-s + 2.21e21·19-s + 1.38e21·20-s + 1.58e21·21-s − 1.05e22·22-s + 3.93e22·23-s − 7.62e21·24-s + 2.32e22·25-s − 1.55e23·26-s − 3.16e23·27-s − 1.22e23·28-s + ⋯
L(s)  = 1  + 1.43·2-s − 1.57·3-s + 1.05·4-s + 0.447·5-s − 2.26·6-s − 0.153·7-s + 0.0815·8-s + 1.48·9-s + 0.641·10-s − 0.522·11-s − 1.66·12-s − 0.486·13-s − 0.219·14-s − 0.704·15-s − 0.939·16-s + 1.63·17-s + 2.12·18-s + 1.76·19-s + 0.472·20-s + 0.241·21-s − 0.749·22-s + 1.33·23-s − 0.128·24-s + 0.200·25-s − 0.698·26-s − 0.763·27-s − 0.162·28-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  =  $(2,\ 5,\ (\ :33/2),\ 1)$
$L(17)$  $\approx$  $2.632504037$
$L(\frac12)$  $\approx$  $2.632504037$
$L(\frac{35}{2})$   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 \( 1 - 1.32e5T + 8.58e9T^{2} \)
3 \( 1 + 1.17e8T + 5.55e15T^{2} \)
7 \( 1 + 1.34e13T + 7.73e27T^{2} \)
11 \( 1 + 7.96e16T + 2.32e34T^{2} \)
13 \( 1 + 1.16e18T + 5.75e36T^{2} \)
17 \( 1 - 3.27e20T + 4.02e40T^{2} \)
19 \( 1 - 2.21e21T + 1.58e42T^{2} \)
23 \( 1 - 3.93e22T + 8.65e44T^{2} \)
29 \( 1 - 1.59e24T + 1.81e48T^{2} \)
31 \( 1 - 2.07e24T + 1.64e49T^{2} \)
37 \( 1 + 4.76e25T + 5.63e51T^{2} \)
41 \( 1 + 6.39e26T + 1.66e53T^{2} \)
43 \( 1 + 2.16e26T + 8.02e53T^{2} \)
47 \( 1 - 7.45e27T + 1.51e55T^{2} \)
53 \( 1 - 1.70e28T + 7.96e56T^{2} \)
59 \( 1 + 5.91e28T + 2.74e58T^{2} \)
61 \( 1 - 2.18e29T + 8.23e58T^{2} \)
67 \( 1 - 2.21e30T + 1.82e60T^{2} \)
71 \( 1 - 2.87e30T + 1.23e61T^{2} \)
73 \( 1 + 3.41e30T + 3.08e61T^{2} \)
79 \( 1 - 5.87e30T + 4.18e62T^{2} \)
83 \( 1 + 2.11e31T + 2.13e63T^{2} \)
89 \( 1 + 2.67e32T + 2.13e64T^{2} \)
97 \( 1 - 1.07e31T + 3.65e65T^{2} \)
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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

−15.70400225918429736187061579168, −13.97838171135516034494130733505, −12.55364572402154932629753025377, −11.68193358755712825584826451446, −10.08690127533644418866853662634, −6.92367029872052807668853523243, −5.52324327154557221332631790441, −5.02992339964104360236355512258, −3.09935720739933656108850924520, −0.890995480028560626908041798196, 0.890995480028560626908041798196, 3.09935720739933656108850924520, 5.02992339964104360236355512258, 5.52324327154557221332631790441, 6.92367029872052807668853523243, 10.08690127533644418866853662634, 11.68193358755712825584826451446, 12.55364572402154932629753025377, 13.97838171135516034494130733505, 15.70400225918429736187061579168

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