Properties

Label 2-8280-1.1-c1-0-105
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 − 0.845·7-s + 4.55·11-s + 5.23·13-s − 4.72·17-s − 5.54·19-s − 23-s + 25-s − 7.06·29-s − 1.83·31-s − 0.845·35-s − 9.93·37-s − 5.71·41-s + 4.78·43-s − 5.54·47-s − 6.28·49-s + 14.4·53-s + 4.55·55-s − 5.06·59-s − 6.22·61-s + 5.23·65-s − 7.85·67-s − 3.71·71-s + 1.23·73-s − 3.85·77-s − 13.7·79-s − 7.50·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.319·7-s + 1.37·11-s + 1.45·13-s − 1.14·17-s − 1.27·19-s − 0.208·23-s + 0.200·25-s − 1.31·29-s − 0.329·31-s − 0.142·35-s − 1.63·37-s − 0.892·41-s + 0.729·43-s − 0.808·47-s − 0.897·49-s + 1.99·53-s + 0.614·55-s − 0.660·59-s − 0.796·61-s + 0.649·65-s − 0.960·67-s − 0.440·71-s + 0.144·73-s − 0.439·77-s − 1.54·79-s − 0.824·83-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 \( 1 + 0.845T + 7T^{2} \)
11 \( 1 - 4.55T + 11T^{2} \)
13 \( 1 - 5.23T + 13T^{2} \)
17 \( 1 + 4.72T + 17T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
29 \( 1 + 7.06T + 29T^{2} \)
31 \( 1 + 1.83T + 31T^{2} \)
37 \( 1 + 9.93T + 37T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 + 5.54T + 47T^{2} \)
53 \( 1 - 14.4T + 53T^{2} \)
59 \( 1 + 5.06T + 59T^{2} \)
61 \( 1 + 6.22T + 61T^{2} \)
67 \( 1 + 7.85T + 67T^{2} \)
71 \( 1 + 3.71T + 71T^{2} \)
73 \( 1 - 1.23T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 7.50T + 83T^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + 2.33T + 97T^{2} \)
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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

−7.24787303145428331997538389264, −6.65852781578630321954787337055, −6.18102268845574781720426677124, −5.58516148744577378532974620790, −4.45375073071408578638415969709, −3.91890786403089708644430837746, −3.22892147656683882581059240387, −1.98113897031378407103206719384, −1.47276135087689712586132679436, 0, 1.47276135087689712586132679436, 1.98113897031378407103206719384, 3.22892147656683882581059240387, 3.91890786403089708644430837746, 4.45375073071408578638415969709, 5.58516148744577378532974620790, 6.18102268845574781720426677124, 6.65852781578630321954787337055, 7.24787303145428331997538389264

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