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Properties

Label 2-5292-1.1-c1-0-37
Degree $2$
Conductor $5292$
Sign $-1$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  − 7·13-s + 8·19-s − 5·25-s + 11·31-s − 37-s − 13·43-s − 61-s + 11·67-s − 10·73-s − 13·79-s − 19·97-s − 7·103-s − 19·109-s + ⋯
L(s)  = 1  − 1.94·13-s + 1.83·19-s − 25-s + 1.97·31-s − 0.164·37-s − 1.98·43-s − 0.128·61-s + 1.34·67-s − 1.17·73-s − 1.46·79-s − 1.92·97-s − 0.689·103-s − 1.81·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−7.79369144481287054638523559764, −7.17630898701742428414985429655, −6.52761080763118846523990874094, −5.46490895426067129351911774496, −5.02127912952499085221407770215, −4.19901402137283051885687562260, −3.12641612058767111677049642157, −2.50141338411949579314311527535, −1.34529262368927578434895691229, 0, 1.34529262368927578434895691229, 2.50141338411949579314311527535, 3.12641612058767111677049642157, 4.19901402137283051885687562260, 5.02127912952499085221407770215, 5.46490895426067129351911774496, 6.52761080763118846523990874094, 7.17630898701742428414985429655, 7.79369144481287054638523559764

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