Properties

Label 2-5292-1.1-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·13-s + 19-s − 5·25-s + 4·31-s − 37-s + 8·43-s + 13·61-s + 11·67-s − 17·73-s − 13·79-s − 5·97-s + 7·103-s + 2·109-s + ⋯
L(s)  = 1  + 1.94·13-s + 0.229·19-s − 25-s + 0.718·31-s − 0.164·37-s + 1.21·43-s + 1.66·61-s + 1.34·67-s − 1.98·73-s − 1.46·79-s − 0.507·97-s + 0.689·103-s + 0.191·109-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: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.185717102\)
\(L(\frac12)\) \(\approx\) \(2.185717102\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.321813700481147925892685842672, −7.51552880521900195309523002532, −6.70078665608916533782840710374, −5.98139804295706451305092616431, −5.49030247404272458638604331312, −4.33573970367631074650890566231, −3.77813458199186956523245005153, −2.91173766719274595341337420733, −1.79680621722010121991203363355, −0.833659174677608845252044473607, 0.833659174677608845252044473607, 1.79680621722010121991203363355, 2.91173766719274595341337420733, 3.77813458199186956523245005153, 4.33573970367631074650890566231, 5.49030247404272458638604331312, 5.98139804295706451305092616431, 6.70078665608916533782840710374, 7.51552880521900195309523002532, 8.321813700481147925892685842672

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