Properties

Label 2-7920-1.1-c1-0-33
Degree $2$
Conductor $7920$
Sign $1$
Analytic cond. $63.2415$
Root an. cond. $7.95245$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3.41·7-s + 11-s − 1.41·13-s − 4.24·17-s + 6.82·19-s + 2.82·23-s + 25-s − 3.65·29-s − 0.828·31-s − 3.41·35-s + 7.65·37-s + 7.65·41-s − 5.07·43-s − 12.4·47-s + 4.65·49-s + 10.4·53-s − 55-s − 3.17·59-s + 3.17·61-s + 1.41·65-s + 8.48·67-s + 6.48·71-s − 7.07·73-s + 3.41·77-s + 16.4·79-s − 1.75·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.29·7-s + 0.301·11-s − 0.392·13-s − 1.02·17-s + 1.56·19-s + 0.589·23-s + 0.200·25-s − 0.679·29-s − 0.148·31-s − 0.577·35-s + 1.25·37-s + 1.19·41-s − 0.773·43-s − 1.82·47-s + 0.665·49-s + 1.44·53-s − 0.134·55-s − 0.412·59-s + 0.406·61-s + 0.175·65-s + 1.03·67-s + 0.769·71-s − 0.827·73-s + 0.389·77-s + 1.85·79-s − 0.192·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(63.2415\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.258505388\)
\(L(\frac12)\) \(\approx\) \(2.258505388\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
good7 \( 1 - 3.41T + 7T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 + 0.828T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 + 5.07T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 3.17T + 59T^{2} \)
61 \( 1 - 3.17T + 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 + 7.07T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + 1.75T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 3.17T + 97T^{2} \)
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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

−7.915267519737862949108166486253, −7.22706338004649759920767944024, −6.62579518065423300431740768651, −5.57556387930043359208140976744, −4.98992372658770539455532675687, −4.39770839082835835666110504710, −3.59228961202000058700130130778, −2.63328086337728497804891513776, −1.72935820272930836841945637350, −0.77476658773066623187586469196, 0.77476658773066623187586469196, 1.72935820272930836841945637350, 2.63328086337728497804891513776, 3.59228961202000058700130130778, 4.39770839082835835666110504710, 4.98992372658770539455532675687, 5.57556387930043359208140976744, 6.62579518065423300431740768651, 7.22706338004649759920767944024, 7.915267519737862949108166486253

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