Properties

Label 2-8280-1.1-c1-0-30
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2.82·7-s + 4.94·11-s + 4.65·13-s + 2.16·17-s − 0.347·19-s − 23-s + 25-s − 1.32·29-s + 10.0·31-s + 2.82·35-s − 5.95·37-s + 4.98·41-s + 1.57·43-s − 13.0·47-s + 1.00·49-s − 5.12·53-s − 4.94·55-s + 12.4·59-s + 10.4·61-s − 4.65·65-s − 12.2·67-s − 4.84·71-s + 12.3·73-s − 13.9·77-s − 8.81·79-s + 14.0·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.06·7-s + 1.49·11-s + 1.29·13-s + 0.525·17-s − 0.0798·19-s − 0.208·23-s + 0.200·25-s − 0.245·29-s + 1.80·31-s + 0.478·35-s − 0.978·37-s + 0.778·41-s + 0.240·43-s − 1.89·47-s + 0.144·49-s − 0.703·53-s − 0.666·55-s + 1.61·59-s + 1.33·61-s − 0.577·65-s − 1.49·67-s − 0.575·71-s + 1.44·73-s − 1.59·77-s − 0.991·79-s + 1.54·83-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: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.911069458\)
\(L(\frac12)\) \(\approx\) \(1.911069458\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 4.94T + 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 - 2.16T + 17T^{2} \)
19 \( 1 + 0.347T + 19T^{2} \)
29 \( 1 + 1.32T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 5.95T + 37T^{2} \)
41 \( 1 - 4.98T + 41T^{2} \)
43 \( 1 - 1.57T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 4.84T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 8.81T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 9.56T + 97T^{2} \)
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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

−7.889596061241644236971422046840, −6.89601636619348690697070695677, −6.44090745853289084907801876701, −6.01777860265393794521555888432, −4.96780896673215715667017249072, −3.98584123027292345588302471878, −3.62697526250809974222098268465, −2.89315226302397280941841447118, −1.58667384438298413357274807031, −0.72027251833946690790423496568, 0.72027251833946690790423496568, 1.58667384438298413357274807031, 2.89315226302397280941841447118, 3.62697526250809974222098268465, 3.98584123027292345588302471878, 4.96780896673215715667017249072, 6.01777860265393794521555888432, 6.44090745853289084907801876701, 6.89601636619348690697070695677, 7.889596061241644236971422046840

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