Properties

Degree 2
Conductor 5077
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.65·2-s − 0.258·3-s + 5.04·4-s − 1.45·5-s + 0.686·6-s − 0.380·7-s − 8.06·8-s − 2.93·9-s + 3.84·10-s + 3.71·11-s − 1.30·12-s − 1.35·13-s + 1.01·14-s + 0.375·15-s + 11.3·16-s + 6.71·17-s + 7.78·18-s + 6.09·19-s − 7.31·20-s + 0.0984·21-s − 9.84·22-s + 7.60·23-s + 2.08·24-s − 2.89·25-s + 3.58·26-s + 1.53·27-s − 1.91·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 0.149·3-s + 2.52·4-s − 0.648·5-s + 0.280·6-s − 0.143·7-s − 2.85·8-s − 0.977·9-s + 1.21·10-s + 1.11·11-s − 0.376·12-s − 0.374·13-s + 0.269·14-s + 0.0968·15-s + 2.83·16-s + 1.62·17-s + 1.83·18-s + 1.39·19-s − 1.63·20-s + 0.0214·21-s − 2.09·22-s + 1.58·23-s + 0.426·24-s − 0.579·25-s + 0.702·26-s + 0.295·27-s − 0.362·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5077\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5077} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5077,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7449863547$
$L(\frac12)$  $\approx$  $0.7449863547$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5077$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 5077$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5077 \( 1+O(T) \)
good2 \( 1 + 2.65T + 2T^{2} \)
3 \( 1 + 0.258T + 3T^{2} \)
5 \( 1 + 1.45T + 5T^{2} \)
7 \( 1 + 0.380T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 + 1.35T + 13T^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 - 6.09T + 19T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 - 2.33T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 - 9.80T + 37T^{2} \)
41 \( 1 - 4.45T + 41T^{2} \)
43 \( 1 - 1.19T + 43T^{2} \)
47 \( 1 - 3.67T + 47T^{2} \)
53 \( 1 - 7.76T + 53T^{2} \)
59 \( 1 + 9.70T + 59T^{2} \)
61 \( 1 + 5.93T + 61T^{2} \)
67 \( 1 + 6.73T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 4.18T + 83T^{2} \)
89 \( 1 - 1.66T + 89T^{2} \)
97 \( 1 - 8.53T + 97T^{2} \)
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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

−8.316046526489725512632256935148, −7.51411556893964741204518515759, −7.31839306195901730006961861485, −6.21526445887267286143353895393, −5.78285625500776149152347522545, −4.53072114995762156043518350759, −3.10201514828723784745308358097, −2.90838465023591052455701049434, −1.29026806704519752890940621006, −0.72952483522846718893432299107, 0.72952483522846718893432299107, 1.29026806704519752890940621006, 2.90838465023591052455701049434, 3.10201514828723784745308358097, 4.53072114995762156043518350759, 5.78285625500776149152347522545, 6.21526445887267286143353895393, 7.31839306195901730006961861485, 7.51411556893964741204518515759, 8.316046526489725512632256935148

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