Properties

Label 2-5096-1.1-c1-0-104
Degree $2$
Conductor $5096$
Sign $-1$
Analytic cond. $40.6917$
Root an. cond. $6.37900$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s + 3·11-s + 13-s − 4·17-s − 2·19-s + 23-s − 5·25-s − 5·27-s + 4·29-s − 9·31-s + 3·33-s + 3·37-s + 39-s + 5·41-s + 4·43-s − 9·47-s − 4·51-s − 4·53-s − 2·57-s − 10·59-s − 5·61-s + 11·67-s + 69-s − 16·71-s − 11·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s + 0.904·11-s + 0.277·13-s − 0.970·17-s − 0.458·19-s + 0.208·23-s − 25-s − 0.962·27-s + 0.742·29-s − 1.61·31-s + 0.522·33-s + 0.493·37-s + 0.160·39-s + 0.780·41-s + 0.609·43-s − 1.31·47-s − 0.560·51-s − 0.549·53-s − 0.264·57-s − 1.30·59-s − 0.640·61-s + 1.34·67-s + 0.120·69-s − 1.89·71-s − 1.28·73-s − 0.577·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5096\)    =    \(2^{3} \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(40.6917\)
Root analytic conductor: \(6.37900\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5096,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.910754372918816081817321150670, −7.26592068191743751180217073693, −6.28624185037441287201175381783, −5.93454858435824926576822829609, −4.79073802262184644752550704108, −4.03852479330444847400169814926, −3.31173661501687262535835297181, −2.39933992770324653977692416071, −1.53934675796141456120027497295, 0, 1.53934675796141456120027497295, 2.39933992770324653977692416071, 3.31173661501687262535835297181, 4.03852479330444847400169814926, 4.79073802262184644752550704108, 5.93454858435824926576822829609, 6.28624185037441287201175381783, 7.26592068191743751180217073693, 7.910754372918816081817321150670

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