Properties

Label 2-7920-1.1-c1-0-14
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 − 2·7-s − 11-s − 1.46·13-s + 1.46·19-s − 6.92·23-s + 25-s − 3.46·29-s − 2.92·31-s − 2·35-s + 8.92·37-s + 3.46·41-s − 8.92·43-s + 6.92·47-s − 3·49-s + 12.9·53-s − 55-s + 6.92·59-s + 2·61-s − 1.46·65-s − 8·67-s − 13.8·71-s + 12.3·73-s + 2·77-s + 13.4·79-s + 15.4·83-s + 12.9·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.301·11-s − 0.406·13-s + 0.335·19-s − 1.44·23-s + 0.200·25-s − 0.643·29-s − 0.525·31-s − 0.338·35-s + 1.46·37-s + 0.541·41-s − 1.36·43-s + 1.01·47-s − 0.428·49-s + 1.77·53-s − 0.134·55-s + 0.901·59-s + 0.256·61-s − 0.181·65-s − 0.977·67-s − 1.64·71-s + 1.45·73-s + 0.227·77-s + 1.51·79-s + 1.69·83-s + 1.37·89-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: $\chi_{7920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542691333\)
\(L(\frac12)\) \(\approx\) \(1.542691333\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 2.92T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.78594280718458666254744357949, −7.16498047217035039555164161798, −6.38229139599154323866638001394, −5.80699869250567149778161706744, −5.18140861575531995253965759736, −4.21769149274301011170668161916, −3.52112900791057735897113384396, −2.60302676456179642555635389026, −1.92464104324579127867055050797, −0.59428919456649012438210385244, 0.59428919456649012438210385244, 1.92464104324579127867055050797, 2.60302676456179642555635389026, 3.52112900791057735897113384396, 4.21769149274301011170668161916, 5.18140861575531995253965759736, 5.80699869250567149778161706744, 6.38229139599154323866638001394, 7.16498047217035039555164161798, 7.78594280718458666254744357949

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