Properties

Label 2-8280-1.1-c1-0-20
Degree $2$
Conductor $8280$
Sign $1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 2·13-s − 2·17-s − 4·19-s − 23-s + 25-s + 8·31-s − 2·35-s + 4·37-s − 4·41-s + 6·43-s − 3·49-s − 2·53-s − 6·59-s − 6·61-s − 2·65-s + 10·67-s + 6·71-s − 14·73-s + 4·79-s + 4·83-s − 2·85-s + 14·89-s + 4·91-s − 4·95-s + 16·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.657·37-s − 0.624·41-s + 0.914·43-s − 3/7·49-s − 0.274·53-s − 0.781·59-s − 0.768·61-s − 0.248·65-s + 1.22·67-s + 0.712·71-s − 1.63·73-s + 0.450·79-s + 0.439·83-s − 0.216·85-s + 1.48·89-s + 0.419·91-s − 0.410·95-s + 1.62·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8280\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(66.1161\)
Root analytic conductor: \(8.13118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542848236\)
\(L(\frac12)\) \(\approx\) \(1.542848236\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−7.80471005093812083810368469264, −7.01285136000994042163306120700, −6.31425505132561024426837340075, −5.99681280955699235256554095887, −4.89463337080128104812434865075, −4.39897332889049803418476310268, −3.40078877612556301385203162821, −2.62826867634854849007154699351, −1.90093926432267635637948436830, −0.58921553515743088733625076828, 0.58921553515743088733625076828, 1.90093926432267635637948436830, 2.62826867634854849007154699351, 3.40078877612556301385203162821, 4.39897332889049803418476310268, 4.89463337080128104812434865075, 5.99681280955699235256554095887, 6.31425505132561024426837340075, 7.01285136000994042163306120700, 7.80471005093812083810368469264

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