Properties

Label 2-7920-1.1-c1-0-89
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 11-s − 6.82·13-s − 1.17·17-s + 2.82·23-s + 25-s − 7.65·29-s + 2·35-s + 3.65·37-s − 6·41-s + 6·43-s − 2.82·47-s − 3·49-s − 0.343·53-s + 55-s − 9.65·59-s + 13.3·61-s − 6.82·65-s + 4.48·67-s − 11.3·71-s − 6.82·73-s + 2·77-s − 4·79-s − 6·83-s − 1.17·85-s − 9.31·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.301·11-s − 1.89·13-s − 0.284·17-s + 0.589·23-s + 0.200·25-s − 1.42·29-s + 0.338·35-s + 0.601·37-s − 0.937·41-s + 0.914·43-s − 0.412·47-s − 0.428·49-s − 0.0471·53-s + 0.134·55-s − 1.25·59-s + 1.70·61-s − 0.846·65-s + 0.547·67-s − 1.34·71-s − 0.799·73-s + 0.227·77-s − 0.450·79-s − 0.658·83-s − 0.127·85-s − 0.987·89-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: \(1\)
Selberg data: \((2,\ 7920,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
good7 \( 1 - 2T + 7T^{2} \)
13 \( 1 + 6.82T + 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 0.343T + 53T^{2} \)
59 \( 1 + 9.65T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 6.82T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−7.39291328395012450288020043006, −6.99783575495615306952749849876, −6.05413838640723643893804521290, −5.27776512102457242011056553286, −4.79834693817511945426182034769, −4.05014062082837923559011409610, −2.93385558158188956855720097389, −2.20958152387170289383865167206, −1.41221950009749839239270445780, 0, 1.41221950009749839239270445780, 2.20958152387170289383865167206, 2.93385558158188956855720097389, 4.05014062082837923559011409610, 4.79834693817511945426182034769, 5.27776512102457242011056553286, 6.05413838640723643893804521290, 6.99783575495615306952749849876, 7.39291328395012450288020043006

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