Properties

Label 2-7920-1.1-c1-0-85
Degree $2$
Conductor $7920$
Sign $-1$
Analytic cond. $63.2415$
Root an. cond. $7.95245$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 4·13-s − 4·17-s − 4·23-s + 25-s + 6·29-s + 8·31-s + 2·35-s − 2·37-s + 10·41-s − 10·43-s − 12·47-s − 3·49-s − 6·53-s − 55-s − 12·59-s + 10·61-s − 4·65-s + 8·67-s + 4·73-s − 2·77-s − 4·79-s + 2·83-s − 4·85-s + 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 0.301·11-s − 1.10·13-s − 0.970·17-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s − 1.52·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.134·55-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 0.977·67-s + 0.468·73-s − 0.227·77-s − 0.450·79-s + 0.219·83-s − 0.433·85-s + 1.05·89-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: yes
Arithmetic: yes
Character: $\chi_{7920} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.67333110756866329874416401978, −6.57327748240581171818887854545, −6.36766165459594369654075552405, −5.08048522522678773409198669875, −4.91305922288279475614074220132, −4.07031386928881137767858771841, −2.87370588881076089291455946719, −2.27520128410229838778949893418, −1.38512320018736524486240508912, 0, 1.38512320018736524486240508912, 2.27520128410229838778949893418, 2.87370588881076089291455946719, 4.07031386928881137767858771841, 4.91305922288279475614074220132, 5.08048522522678773409198669875, 6.36766165459594369654075552405, 6.57327748240581171818887854545, 7.67333110756866329874416401978

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