Properties

Label 2-8280-1.1-c1-0-100
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 + 4.86·7-s − 4.77·11-s + 2.77·13-s + 0.636·17-s − 3.50·19-s + 23-s + 25-s − 7.36·29-s − 5.15·31-s − 4.86·35-s + 2.86·37-s + 6.19·41-s − 8.28·43-s − 7.50·47-s + 16.6·49-s + 4.91·53-s + 4.77·55-s + 8.65·59-s − 15.0·61-s − 2.77·65-s − 13.4·67-s + 11.0·71-s − 15.2·73-s − 23.2·77-s − 1.45·79-s − 6.63·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.83·7-s − 1.44·11-s + 0.770·13-s + 0.154·17-s − 0.804·19-s + 0.208·23-s + 0.200·25-s − 1.36·29-s − 0.925·31-s − 0.822·35-s + 0.471·37-s + 0.967·41-s − 1.26·43-s − 1.09·47-s + 2.38·49-s + 0.675·53-s + 0.644·55-s + 1.12·59-s − 1.92·61-s − 0.344·65-s − 1.64·67-s + 1.31·71-s − 1.78·73-s − 2.65·77-s − 0.163·79-s − 0.728·83-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 \( 1 - 4.86T + 7T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 - 0.636T + 17T^{2} \)
19 \( 1 + 3.50T + 19T^{2} \)
29 \( 1 + 7.36T + 29T^{2} \)
31 \( 1 + 5.15T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 - 6.19T + 41T^{2} \)
43 \( 1 + 8.28T + 43T^{2} \)
47 \( 1 + 7.50T + 47T^{2} \)
53 \( 1 - 4.91T + 53T^{2} \)
59 \( 1 - 8.65T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 1.45T + 79T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + 9.29T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−7.66238063779520186934663592873, −6.99996491943777037569498342216, −5.82571331685248834173087770793, −5.38574451515301161649416186683, −4.62799841925315499331988688763, −4.06778004704683617759735997552, −3.06325743169235666847856123150, −2.09274549234136310463423846564, −1.38770518968184600851634418459, 0, 1.38770518968184600851634418459, 2.09274549234136310463423846564, 3.06325743169235666847856123150, 4.06778004704683617759735997552, 4.62799841925315499331988688763, 5.38574451515301161649416186683, 5.82571331685248834173087770793, 6.99996491943777037569498342216, 7.66238063779520186934663592873

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