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  − 2.49·2-s + 1.36·3-s + 4.21·4-s − 1.81·5-s − 3.39·6-s + 0.512·7-s − 5.50·8-s − 1.14·9-s + 4.52·10-s − 3.00·11-s + 5.72·12-s − 2.35·13-s − 1.27·14-s − 2.46·15-s + 5.30·16-s + 3.40·17-s + 2.86·18-s − 3.31·19-s − 7.63·20-s + 0.697·21-s + 7.49·22-s − 4.63·23-s − 7.49·24-s − 1.70·25-s + 5.87·26-s − 5.64·27-s + 2.15·28-s + ⋯
L(s)  = 1  − 1.76·2-s + 0.785·3-s + 2.10·4-s − 0.811·5-s − 1.38·6-s + 0.193·7-s − 1.94·8-s − 0.382·9-s + 1.42·10-s − 0.907·11-s + 1.65·12-s − 0.654·13-s − 0.341·14-s − 0.637·15-s + 1.32·16-s + 0.824·17-s + 0.674·18-s − 0.760·19-s − 1.70·20-s + 0.152·21-s + 1.59·22-s − 0.965·23-s − 1.52·24-s − 0.341·25-s + 1.15·26-s − 1.08·27-s + 0.407·28-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  =  $(2,\ 5077,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4174199700$
$L(\frac12)$  $\approx$  $0.4174199700$
$L(\frac{3}{2})$   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 \( 1 + 2.49T + 2T^{2} \)
3 \( 1 - 1.36T + 3T^{2} \)
5 \( 1 + 1.81T + 5T^{2} \)
7 \( 1 - 0.512T + 7T^{2} \)
11 \( 1 + 3.00T + 11T^{2} \)
13 \( 1 + 2.35T + 13T^{2} \)
17 \( 1 - 3.40T + 17T^{2} \)
19 \( 1 + 3.31T + 19T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 - 4.84T + 29T^{2} \)
31 \( 1 + 7.08T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 - 9.86T + 41T^{2} \)
43 \( 1 - 2.36T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 6.73T + 53T^{2} \)
59 \( 1 + 3.76T + 59T^{2} \)
61 \( 1 + 5.85T + 61T^{2} \)
67 \( 1 + 7.69T + 67T^{2} \)
71 \( 1 - 0.317T + 71T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 0.399T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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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

−8.031684340445378378972828713652, −7.82512506649653026164140233651, −7.46784205343188633280610535504, −6.36353069248634546978177929534, −5.53950666402215488991546037273, −4.36904895488981269471798671628, −3.34643766192976789360006374368, −2.54168597833795212303824155424, −1.86097652113631610136243292313, −0.41709347599443254848941830598, 0.41709347599443254848941830598, 1.86097652113631610136243292313, 2.54168597833795212303824155424, 3.34643766192976789360006374368, 4.36904895488981269471798671628, 5.53950666402215488991546037273, 6.36353069248634546978177929534, 7.46784205343188633280610535504, 7.82512506649653026164140233651, 8.031684340445378378972828713652

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