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  − 1.82e4·2-s − 3.67e7·3-s − 8.25e9·4-s + 1.52e11·5-s + 6.70e11·6-s − 1.51e13·7-s + 3.07e14·8-s − 4.20e15·9-s − 2.78e15·10-s − 2.61e17·11-s + 3.03e17·12-s − 3.70e17·13-s + 2.76e17·14-s − 5.60e18·15-s + 6.53e19·16-s − 3.71e20·17-s + 7.68e19·18-s + 1.42e21·19-s − 1.25e21·20-s + 5.57e20·21-s + 4.77e21·22-s − 3.26e22·23-s − 1.13e22·24-s + 2.32e22·25-s + 6.76e21·26-s + 3.58e23·27-s + 1.25e23·28-s + ⋯
L(s)  = 1  − 0.197·2-s − 0.492·3-s − 0.961·4-s + 0.447·5-s + 0.0970·6-s − 0.172·7-s + 0.386·8-s − 0.757·9-s − 0.0881·10-s − 1.71·11-s + 0.473·12-s − 0.154·13-s + 0.0339·14-s − 0.220·15-s + 0.885·16-s − 1.85·17-s + 0.149·18-s + 1.13·19-s − 0.429·20-s + 0.0850·21-s + 0.338·22-s − 1.11·23-s − 0.190·24-s + 0.200·25-s + 0.0304·26-s + 0.865·27-s + 0.165·28-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  =  $(2,\ 5,\ (\ :33/2),\ 1)$
$L(17)$  $\approx$  $0.5716579424$
$L(\frac12)$  $\approx$  $0.5716579424$
$L(\frac{35}{2})$   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 \( 1 + 1.82e4T + 8.58e9T^{2} \)
3 \( 1 + 3.67e7T + 5.55e15T^{2} \)
7 \( 1 + 1.51e13T + 7.73e27T^{2} \)
11 \( 1 + 2.61e17T + 2.32e34T^{2} \)
13 \( 1 + 3.70e17T + 5.75e36T^{2} \)
17 \( 1 + 3.71e20T + 4.02e40T^{2} \)
19 \( 1 - 1.42e21T + 1.58e42T^{2} \)
23 \( 1 + 3.26e22T + 8.65e44T^{2} \)
29 \( 1 - 6.41e23T + 1.81e48T^{2} \)
31 \( 1 + 1.44e23T + 1.64e49T^{2} \)
37 \( 1 - 1.34e26T + 5.63e51T^{2} \)
41 \( 1 - 6.37e26T + 1.66e53T^{2} \)
43 \( 1 - 3.37e26T + 8.02e53T^{2} \)
47 \( 1 - 3.12e27T + 1.51e55T^{2} \)
53 \( 1 + 4.06e28T + 7.96e56T^{2} \)
59 \( 1 + 8.77e28T + 2.74e58T^{2} \)
61 \( 1 + 1.84e28T + 8.23e58T^{2} \)
67 \( 1 - 7.69e29T + 1.82e60T^{2} \)
71 \( 1 - 2.72e30T + 1.23e61T^{2} \)
73 \( 1 - 4.44e30T + 3.08e61T^{2} \)
79 \( 1 + 1.16e31T + 4.18e62T^{2} \)
83 \( 1 + 4.84e31T + 2.13e63T^{2} \)
89 \( 1 - 1.18e32T + 2.13e64T^{2} \)
97 \( 1 + 2.71e31T + 3.65e65T^{2} \)
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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

−15.96752080440111405977631773776, −13.99158390602329051340182936035, −12.87579028573442615598843893884, −10.92491670184592227047688102834, −9.512096116361041587481599256465, −8.014651417893189671845307788602, −5.86910312870096593688730688356, −4.69396255265465087321603106371, −2.59899542873453747807745214743, −0.46818766107997323846354214956, 0.46818766107997323846354214956, 2.59899542873453747807745214743, 4.69396255265465087321603106371, 5.86910312870096593688730688356, 8.014651417893189671845307788602, 9.512096116361041587481599256465, 10.92491670184592227047688102834, 12.87579028573442615598843893884, 13.99158390602329051340182936035, 15.96752080440111405977631773776

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