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  − 2.46·2-s − 2.68·3-s + 4.05·4-s + 2.96·5-s + 6.59·6-s + 0.509·7-s − 5.06·8-s + 4.19·9-s − 7.29·10-s + 5.31·11-s − 10.8·12-s + 5.60·13-s − 1.25·14-s − 7.94·15-s + 4.34·16-s + 2.47·17-s − 10.3·18-s − 0.319·19-s + 12.0·20-s − 1.36·21-s − 13.0·22-s + 2.93·23-s + 13.5·24-s + 3.78·25-s − 13.7·26-s − 3.19·27-s + 2.06·28-s + ⋯
L(s)  = 1  − 1.74·2-s − 1.54·3-s + 2.02·4-s + 1.32·5-s + 2.69·6-s + 0.192·7-s − 1.78·8-s + 1.39·9-s − 2.30·10-s + 1.60·11-s − 3.14·12-s + 1.55·13-s − 0.335·14-s − 2.05·15-s + 1.08·16-s + 0.599·17-s − 2.43·18-s − 0.0733·19-s + 2.68·20-s − 0.298·21-s − 2.78·22-s + 0.612·23-s + 2.77·24-s + 0.757·25-s − 2.70·26-s − 0.615·27-s + 0.390·28-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  =  $(2,\ 5077,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9633387543$
$L(\frac12)$  $\approx$  $0.9633387543$
$L(\frac{3}{2})$   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 \( 1 + 2.46T + 2T^{2} \)
3 \( 1 + 2.68T + 3T^{2} \)
5 \( 1 - 2.96T + 5T^{2} \)
7 \( 1 - 0.509T + 7T^{2} \)
11 \( 1 - 5.31T + 11T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 + 0.319T + 19T^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
31 \( 1 - 7.77T + 31T^{2} \)
37 \( 1 - 9.69T + 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 - 2.69T + 43T^{2} \)
47 \( 1 - 1.32T + 47T^{2} \)
53 \( 1 - 3.72T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 2.43T + 61T^{2} \)
67 \( 1 - 9.21T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + 6.94T + 73T^{2} \)
79 \( 1 + 9.50T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.501479692071198381886509087975, −7.47191159182041294530245936868, −6.58014471180274492247370583491, −6.27231053676058098614851704361, −5.84322342804376504133439621501, −4.83037472368913930641247738800, −3.66164566187486925424122275309, −2.20342573806860051277105059413, −1.16680563723700876545470665618, −1.00851312280412574868396732162, 1.00851312280412574868396732162, 1.16680563723700876545470665618, 2.20342573806860051277105059413, 3.66164566187486925424122275309, 4.83037472368913930641247738800, 5.84322342804376504133439621501, 6.27231053676058098614851704361, 6.58014471180274492247370583491, 7.47191159182041294530245936868, 8.501479692071198381886509087975

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