Properties

Degree 2
Conductor 5077
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 4·7-s + 6·9-s + 8·10-s − 6·11-s − 6·12-s − 4·13-s + 8·14-s + 12·15-s − 4·16-s − 4·17-s − 12·18-s − 7·19-s − 8·20-s + 12·21-s + 12·22-s − 6·23-s + 11·25-s + 8·26-s − 9·27-s − 8·28-s − 6·29-s − 24·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 1.78·5-s + 2.44·6-s − 1.51·7-s + 2·9-s + 2.52·10-s − 1.80·11-s − 1.73·12-s − 1.10·13-s + 2.13·14-s + 3.09·15-s − 16-s − 0.970·17-s − 2.82·18-s − 1.60·19-s − 1.78·20-s + 2.61·21-s + 2.55·22-s − 1.25·23-s + 11/5·25-s + 1.56·26-s − 1.73·27-s − 1.51·28-s − 1.11·29-s − 4.38·30-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  =  3
Selberg data  =  $(2,\ 5077,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5077 \( 1+O(T) \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.34281497953964814691434021056, −6.62250461340770678139848771792, −6.01192275298639519014642522249, −4.75443151596340586415163559397, −4.47055151331009795091782387950, −3.26244355597875746635580364386, −2.05247285847993976968922276314, 0, 0, 0, 2.05247285847993976968922276314, 3.26244355597875746635580364386, 4.47055151331009795091782387950, 4.75443151596340586415163559397, 6.01192275298639519014642522249, 6.62250461340770678139848771792, 7.34281497953964814691434021056

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