Properties

Label 2-7920-1.1-c1-0-65
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 − 1.56·7-s + 11-s + 2·13-s − 3.56·17-s + 1.56·19-s − 3.12·23-s + 25-s + 2.68·29-s − 2.43·31-s + 1.56·35-s + 6.68·37-s − 2·41-s + 6.24·43-s − 4.87·47-s − 4.56·49-s − 0.438·53-s − 55-s − 7.12·59-s + 14.6·61-s − 2·65-s + 10.2·67-s − 8.68·71-s − 2·73-s − 1.56·77-s − 9.36·79-s − 3.12·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.590·7-s + 0.301·11-s + 0.554·13-s − 0.863·17-s + 0.358·19-s − 0.651·23-s + 0.200·25-s + 0.498·29-s − 0.437·31-s + 0.263·35-s + 1.09·37-s − 0.312·41-s + 0.952·43-s − 0.711·47-s − 0.651·49-s − 0.0602·53-s − 0.134·55-s − 0.927·59-s + 1.88·61-s − 0.248·65-s + 1.25·67-s − 1.03·71-s − 0.234·73-s − 0.177·77-s − 1.05·79-s − 0.342·83-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: \(1\)
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 \( 1 + 1.56T + 7T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 - 2.68T + 29T^{2} \)
31 \( 1 + 2.43T + 31T^{2} \)
37 \( 1 - 6.68T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 + 4.87T + 47T^{2} \)
53 \( 1 + 0.438T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 8.68T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 9.36T + 79T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 - 8.43T + 89T^{2} \)
97 \( 1 + 1.12T + 97T^{2} \)
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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

−7.49239849729328195465932937746, −6.72633103393073266004091860030, −6.24890522141297399451946130037, −5.46872103272161474476497025779, −4.51536497150318530782591768223, −3.92761848324211068144625572683, −3.17931771598565961749959093966, −2.31005459100045372567236358959, −1.17787551093557067709920220950, 0, 1.17787551093557067709920220950, 2.31005459100045372567236358959, 3.17931771598565961749959093966, 3.92761848324211068144625572683, 4.51536497150318530782591768223, 5.46872103272161474476497025779, 6.24890522141297399451946130037, 6.72633103393073266004091860030, 7.49239849729328195465932937746

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