Properties

Degree 2
Conductor 5
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.28·2-s + 79.7·3-s − 126.·4-s − 125·5-s + 102.·6-s − 538.·7-s − 326.·8-s + 4.17e3·9-s − 160.·10-s − 1.21e3·11-s − 1.00e4·12-s + 7.07e3·13-s − 690.·14-s − 9.96e3·15-s + 1.57e4·16-s − 3.34e3·17-s + 5.34e3·18-s + 2.21e4·19-s + 1.57e4·20-s − 4.29e4·21-s − 1.55e3·22-s − 5.85e4·23-s − 2.60e4·24-s + 1.56e4·25-s + 9.06e3·26-s + 1.58e5·27-s + 6.80e4·28-s + ⋯
L(s)  = 1  + 0.113·2-s + 1.70·3-s − 0.987·4-s − 0.447·5-s + 0.193·6-s − 0.593·7-s − 0.225·8-s + 1.90·9-s − 0.0506·10-s − 0.275·11-s − 1.68·12-s + 0.892·13-s − 0.0672·14-s − 0.762·15-s + 0.961·16-s − 0.165·17-s + 0.216·18-s + 0.741·19-s + 0.441·20-s − 1.01·21-s − 0.0311·22-s − 1.00·23-s − 0.384·24-s + 0.199·25-s + 0.101·26-s + 1.54·27-s + 0.585·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{5} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $1.46766$
$L(\frac12)$  $\approx$  $1.46766$
$L(\frac{9}{2})$   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 \( 1 + 125T \)
good2 \( 1 - 1.28T + 128T^{2} \)
3 \( 1 - 79.7T + 2.18e3T^{2} \)
7 \( 1 + 538.T + 8.23e5T^{2} \)
11 \( 1 + 1.21e3T + 1.94e7T^{2} \)
13 \( 1 - 7.07e3T + 6.27e7T^{2} \)
17 \( 1 + 3.34e3T + 4.10e8T^{2} \)
19 \( 1 - 2.21e4T + 8.93e8T^{2} \)
23 \( 1 + 5.85e4T + 3.40e9T^{2} \)
29 \( 1 + 2.06e5T + 1.72e10T^{2} \)
31 \( 1 - 1.77e5T + 2.75e10T^{2} \)
37 \( 1 + 2.84e5T + 9.49e10T^{2} \)
41 \( 1 - 6.27e5T + 1.94e11T^{2} \)
43 \( 1 + 1.64e5T + 2.71e11T^{2} \)
47 \( 1 + 4.49e5T + 5.06e11T^{2} \)
53 \( 1 + 7.30e5T + 1.17e12T^{2} \)
59 \( 1 - 1.42e6T + 2.48e12T^{2} \)
61 \( 1 + 2.66e5T + 3.14e12T^{2} \)
67 \( 1 - 2.95e6T + 6.06e12T^{2} \)
71 \( 1 - 9.21e5T + 9.09e12T^{2} \)
73 \( 1 - 4.25e6T + 1.10e13T^{2} \)
79 \( 1 - 6.28e6T + 1.92e13T^{2} \)
83 \( 1 + 9.17e6T + 2.71e13T^{2} \)
89 \( 1 - 2.42e5T + 4.42e13T^{2} \)
97 \( 1 + 2.59e6T + 8.07e13T^{2} \)
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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

−22.60501002348992296283196614035, −20.88966920745470466595519261826, −19.52897156300637550527526803721, −18.41503997148447739979704080654, −15.70441842117171651071931414555, −14.15821035961646171329220384062, −13.04321206354133836289379117522, −9.539264410826295415867260441329, −8.132385337579659451415013951776, −3.63715009395516665553552623709, 3.63715009395516665553552623709, 8.132385337579659451415013951776, 9.539264410826295415867260441329, 13.04321206354133836289379117522, 14.15821035961646171329220384062, 15.70441842117171651071931414555, 18.41503997148447739979704080654, 19.52897156300637550527526803721, 20.88966920745470466595519261826, 22.60501002348992296283196614035

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