Properties

Label 2-8280-1.1-c1-0-108
Degree $2$
Conductor $8280$
Sign $-1$
Analytic cond. $66.1161$
Root an. cond. $8.13118$
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 + 4.46·7-s − 3.39·11-s − 4.70·13-s + 6.54·17-s − 6.22·19-s − 23-s + 25-s − 0.448·29-s − 5.15·31-s + 4.46·35-s − 5.98·37-s − 3.07·41-s − 4.47·43-s − 6.22·47-s + 12.9·49-s + 2.59·53-s − 3.39·55-s + 1.55·59-s − 4.91·61-s − 4.70·65-s + 6.08·67-s − 1.07·71-s − 8.70·73-s − 15.1·77-s − 13.1·79-s + 13.0·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.68·7-s − 1.02·11-s − 1.30·13-s + 1.58·17-s − 1.42·19-s − 0.208·23-s + 0.200·25-s − 0.0832·29-s − 0.925·31-s + 0.754·35-s − 0.984·37-s − 0.479·41-s − 0.683·43-s − 0.908·47-s + 1.84·49-s + 0.355·53-s − 0.457·55-s + 0.201·59-s − 0.629·61-s − 0.583·65-s + 0.743·67-s − 0.127·71-s − 1.01·73-s − 1.72·77-s − 1.47·79-s + 1.43·83-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: no
Arithmetic: yes
Character: $\chi_{8280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 - 4.46T + 7T^{2} \)
11 \( 1 + 3.39T + 11T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 - 6.54T + 17T^{2} \)
19 \( 1 + 6.22T + 19T^{2} \)
29 \( 1 + 0.448T + 29T^{2} \)
31 \( 1 + 5.15T + 31T^{2} \)
37 \( 1 + 5.98T + 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 6.22T + 47T^{2} \)
53 \( 1 - 2.59T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 + 4.91T + 61T^{2} \)
67 \( 1 - 6.08T + 67T^{2} \)
71 \( 1 + 1.07T + 71T^{2} \)
73 \( 1 + 8.70T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 - 4.30T + 97T^{2} \)
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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

−7.54776360961254981339061880386, −6.96618433448059809220273629319, −5.86594979083031180282332150257, −5.18223676055848971802500436494, −4.94835707725241661566473900987, −4.01395501655955107898909581193, −2.91821318284787962862050081776, −2.08902931574829899628105974442, −1.50350791414627311153038069234, 0, 1.50350791414627311153038069234, 2.08902931574829899628105974442, 2.91821318284787962862050081776, 4.01395501655955107898909581193, 4.94835707725241661566473900987, 5.18223676055848971802500436494, 5.86594979083031180282332150257, 6.96618433448059809220273629319, 7.54776360961254981339061880386

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