Properties

Label 2-8280-1.1-c1-0-49
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 + 4·7-s + 6·11-s − 4·13-s − 2·17-s − 23-s + 25-s − 8·29-s + 8·31-s + 4·35-s − 6·37-s + 2·41-s + 6·43-s + 12·47-s + 9·49-s − 6·53-s + 6·55-s − 4·59-s + 2·61-s − 4·65-s + 6·67-s + 6·71-s + 14·73-s + 24·77-s − 2·85-s − 12·89-s − 16·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 1.80·11-s − 1.10·13-s − 0.485·17-s − 0.208·23-s + 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.312·41-s + 0.914·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.809·55-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.733·67-s + 0.712·71-s + 1.63·73-s + 2.73·77-s − 0.216·85-s − 1.27·89-s − 1.67·91-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.075956968\)
\(L(\frac12)\) \(\approx\) \(3.075956968\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.73073234657050150623341480445, −7.17710859072532218036879145389, −6.44766158430502723771433890053, −5.71128825270245073853630312862, −4.92089415833609027730817093822, −4.37923231849502679376550358717, −3.66292423675929440929574123162, −2.36785677867917375107863105079, −1.81702010177320928428099208215, −0.911516956997658826024212794738, 0.911516956997658826024212794738, 1.81702010177320928428099208215, 2.36785677867917375107863105079, 3.66292423675929440929574123162, 4.37923231849502679376550358717, 4.92089415833609027730817093822, 5.71128825270245073853630312862, 6.44766158430502723771433890053, 7.17710859072532218036879145389, 7.73073234657050150623341480445

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