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  − 2.71·2-s − 1.56·3-s + 5.35·4-s + 1.41·5-s + 4.23·6-s − 1.24·7-s − 9.11·8-s − 0.564·9-s − 3.84·10-s − 1.09·11-s − 8.36·12-s − 1.70·13-s + 3.36·14-s − 2.21·15-s + 14.0·16-s + 0.0690·17-s + 1.53·18-s − 5.12·19-s + 7.60·20-s + 1.93·21-s + 2.95·22-s + 5.50·23-s + 14.2·24-s − 2.98·25-s + 4.62·26-s + 5.56·27-s − 6.64·28-s + ⋯
L(s)  = 1  − 1.91·2-s − 0.901·3-s + 2.67·4-s + 0.634·5-s + 1.72·6-s − 0.468·7-s − 3.22·8-s − 0.188·9-s − 1.21·10-s − 0.328·11-s − 2.41·12-s − 0.472·13-s + 0.899·14-s − 0.571·15-s + 3.50·16-s + 0.0167·17-s + 0.360·18-s − 1.17·19-s + 1.69·20-s + 0.422·21-s + 0.630·22-s + 1.14·23-s + 2.90·24-s − 0.597·25-s + 0.906·26-s + 1.07·27-s − 1.25·28-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  =  $(2,\ 5077,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2346231230$
$L(\frac12)$  $\approx$  $0.2346231230$
$L(\frac{3}{2})$   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 \( 1 + 2.71T + 2T^{2} \)
3 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + 1.09T + 11T^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
17 \( 1 - 0.0690T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 - 5.50T + 23T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 + 1.98T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 2.16T + 41T^{2} \)
43 \( 1 + 7.04T + 43T^{2} \)
47 \( 1 + 6.34T + 47T^{2} \)
53 \( 1 + 9.99T + 53T^{2} \)
59 \( 1 + 7.58T + 59T^{2} \)
61 \( 1 + 7.37T + 61T^{2} \)
67 \( 1 - 7.07T + 67T^{2} \)
71 \( 1 - 5.26T + 71T^{2} \)
73 \( 1 - 8.99T + 73T^{2} \)
79 \( 1 - 4.05T + 79T^{2} \)
83 \( 1 - 1.64T + 83T^{2} \)
89 \( 1 - 8.31T + 89T^{2} \)
97 \( 1 + 7.87T + 97T^{2} \)
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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

−8.183759360186699542350233229144, −7.76182323374325760290745347784, −6.61344933795254148249358059353, −6.49869475369615393875965083595, −5.70128276581111558025982709549, −4.86755373676768049140597690312, −3.26551089087587380502126839833, −2.43556731381829002946412980925, −1.58523169219992779609272723961, −0.36259107103742167078030915259, 0.36259107103742167078030915259, 1.58523169219992779609272723961, 2.43556731381829002946412980925, 3.26551089087587380502126839833, 4.86755373676768049140597690312, 5.70128276581111558025982709549, 6.49869475369615393875965083595, 6.61344933795254148249358059353, 7.76182323374325760290745347784, 8.183759360186699542350233229144

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