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Properties

Label 2-5292-1.1-c1-0-38
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  − 2·13-s + 19-s − 5·25-s + 7·31-s − 10·37-s + 5·43-s + 61-s − 16·67-s − 17·73-s − 4·79-s + 19·97-s − 20·103-s + 17·109-s + ⋯
L(s)  = 1  − 0.554·13-s + 0.229·19-s − 25-s + 1.25·31-s − 1.64·37-s + 0.762·43-s + 0.128·61-s − 1.95·67-s − 1.98·73-s − 0.450·79-s + 1.92·97-s − 1.97·103-s + 1.62·109-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
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

−7.72199306134393138306407573998, −7.24594222552258881123101900522, −6.36715281735529416169887362218, −5.69450547655071894939204656548, −4.88352720543090562411837467317, −4.16604500921602563768270826894, −3.24453818207954994047258424100, −2.40070330307379079292747947409, −1.37942178505304104458472745007, 0, 1.37942178505304104458472745007, 2.40070330307379079292747947409, 3.24453818207954994047258424100, 4.16604500921602563768270826894, 4.88352720543090562411837467317, 5.69450547655071894939204656548, 6.36715281735529416169887362218, 7.24594222552258881123101900522, 7.72199306134393138306407573998

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