Properties

Label 2-8280-1.1-c1-0-11
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 − 0.944·7-s − 5.62·11-s + 4.42·13-s − 3.41·17-s − 2.35·19-s + 23-s + 25-s − 8.57·29-s − 8.99·31-s + 0.944·35-s + 1.05·37-s + 9.35·41-s + 4.78·43-s − 2.35·47-s − 6.10·49-s + 11.4·53-s + 5.62·55-s − 13.2·59-s + 11.6·61-s − 4.42·65-s − 8.93·67-s + 4.68·71-s + 1.53·73-s + 5.31·77-s + 16.7·79-s + 5.41·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.357·7-s − 1.69·11-s + 1.22·13-s − 0.828·17-s − 0.541·19-s + 0.208·23-s + 0.200·25-s − 1.59·29-s − 1.61·31-s + 0.159·35-s + 0.173·37-s + 1.46·41-s + 0.729·43-s − 0.344·47-s − 0.872·49-s + 1.57·53-s + 0.758·55-s − 1.72·59-s + 1.49·61-s − 0.548·65-s − 1.09·67-s + 0.555·71-s + 0.179·73-s + 0.605·77-s + 1.88·79-s + 0.594·83-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

\(L(1)\) \(\approx\) \(0.9568123642\)
\(L(\frac12)\) \(\approx\) \(0.9568123642\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.944T + 7T^{2} \)
11 \( 1 + 5.62T + 11T^{2} \)
13 \( 1 - 4.42T + 13T^{2} \)
17 \( 1 + 3.41T + 17T^{2} \)
19 \( 1 + 2.35T + 19T^{2} \)
29 \( 1 + 8.57T + 29T^{2} \)
31 \( 1 + 8.99T + 31T^{2} \)
37 \( 1 - 1.05T + 37T^{2} \)
41 \( 1 - 9.35T + 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 8.93T + 67T^{2} \)
71 \( 1 - 4.68T + 71T^{2} \)
73 \( 1 - 1.53T + 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 - 5.41T + 83T^{2} \)
89 \( 1 + 18.8T + 89T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−7.68031560341228996209230195269, −7.32756715758634408626039786937, −6.35936409030260970362588050183, −5.74099635003379820536201752380, −5.08993238855482861556521728513, −4.14261257908554026966187488337, −3.57400840190912942996989389609, −2.66586559047192184633676977326, −1.87224405904429818286111278500, −0.45927436485173521828108752384, 0.45927436485173521828108752384, 1.87224405904429818286111278500, 2.66586559047192184633676977326, 3.57400840190912942996989389609, 4.14261257908554026966187488337, 5.08993238855482861556521728513, 5.74099635003379820536201752380, 6.35936409030260970362588050183, 7.32756715758634408626039786937, 7.68031560341228996209230195269

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