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  + 1.38e5·2-s − 1.45e7·3-s + 1.04e10·4-s − 1.52e11·5-s − 2.00e12·6-s + 3.75e13·7-s + 2.61e14·8-s − 5.34e15·9-s − 2.10e16·10-s − 7.18e16·11-s − 1.52e17·12-s + 1.13e18·13-s + 5.18e18·14-s + 2.21e18·15-s − 5.39e19·16-s − 1.70e20·17-s − 7.38e20·18-s − 8.48e20·19-s − 1.59e21·20-s − 5.45e20·21-s − 9.92e21·22-s − 3.95e22·23-s − 3.79e21·24-s + 2.32e22·25-s + 1.56e23·26-s + 1.58e23·27-s + 3.93e23·28-s + ⋯
L(s)  = 1  + 1.49·2-s − 0.194·3-s + 1.22·4-s − 0.447·5-s − 0.290·6-s + 0.427·7-s + 0.328·8-s − 0.962·9-s − 0.666·10-s − 0.471·11-s − 0.237·12-s + 0.471·13-s + 0.636·14-s + 0.0871·15-s − 0.731·16-s − 0.851·17-s − 1.43·18-s − 0.674·19-s − 0.545·20-s − 0.0832·21-s − 0.703·22-s − 1.34·23-s − 0.0639·24-s + 0.200·25-s + 0.703·26-s + 0.382·27-s + 0.521·28-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 \( 1 - 1.38e5T + 8.58e9T^{2} \)
3 \( 1 + 1.45e7T + 5.55e15T^{2} \)
7 \( 1 - 3.75e13T + 7.73e27T^{2} \)
11 \( 1 + 7.18e16T + 2.32e34T^{2} \)
13 \( 1 - 1.13e18T + 5.75e36T^{2} \)
17 \( 1 + 1.70e20T + 4.02e40T^{2} \)
19 \( 1 + 8.48e20T + 1.58e42T^{2} \)
23 \( 1 + 3.95e22T + 8.65e44T^{2} \)
29 \( 1 - 9.30e23T + 1.81e48T^{2} \)
31 \( 1 + 8.60e23T + 1.64e49T^{2} \)
37 \( 1 + 5.07e25T + 5.63e51T^{2} \)
41 \( 1 + 1.78e26T + 1.66e53T^{2} \)
43 \( 1 - 8.10e26T + 8.02e53T^{2} \)
47 \( 1 + 5.64e27T + 1.51e55T^{2} \)
53 \( 1 - 5.15e28T + 7.96e56T^{2} \)
59 \( 1 - 1.40e29T + 2.74e58T^{2} \)
61 \( 1 + 8.28e28T + 8.23e58T^{2} \)
67 \( 1 - 2.02e29T + 1.82e60T^{2} \)
71 \( 1 - 6.89e30T + 1.23e61T^{2} \)
73 \( 1 + 3.84e30T + 3.08e61T^{2} \)
79 \( 1 - 9.36e30T + 4.18e62T^{2} \)
83 \( 1 - 7.23e31T + 2.13e63T^{2} \)
89 \( 1 - 4.37e31T + 2.13e64T^{2} \)
97 \( 1 + 1.02e33T + 3.65e65T^{2} \)
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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

−14.73946393905834999194974979725, −13.54914939973821801423964083512, −12.09839664115592827903175529120, −11.00410138318081796199288458716, −8.400472955148589293271965183543, −6.39248164769554864033054971017, −5.10432073364808440396615863509, −3.82382516576215466991583202151, −2.35899316743169770198591914298, 0, 2.35899316743169770198591914298, 3.82382516576215466991583202151, 5.10432073364808440396615863509, 6.39248164769554864033054971017, 8.400472955148589293271965183543, 11.00410138318081796199288458716, 12.09839664115592827903175529120, 13.54914939973821801423964083512, 14.73946393905834999194974979725

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