Properties

Label 2-8280-1.1-c1-0-106
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 + 3.20·7-s + 5.11·11-s + 0.597·13-s − 5.98·17-s − 3.11·19-s − 23-s + 25-s − 4.64·29-s + 3.39·31-s − 3.20·35-s − 6.51·37-s − 11.4·41-s − 3.71·43-s + 1.80·47-s + 3.28·49-s − 8.13·53-s − 5.11·55-s + 3.80·59-s − 7.07·61-s − 0.597·65-s + 0.0987·67-s − 9.51·71-s − 7.41·73-s + 16.3·77-s − 5.96·79-s + 4.55·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.21·7-s + 1.54·11-s + 0.165·13-s − 1.45·17-s − 0.714·19-s − 0.208·23-s + 0.200·25-s − 0.862·29-s + 0.608·31-s − 0.542·35-s − 1.07·37-s − 1.79·41-s − 0.565·43-s + 0.263·47-s + 0.468·49-s − 1.11·53-s − 0.689·55-s + 0.495·59-s − 0.905·61-s − 0.0741·65-s + 0.0120·67-s − 1.12·71-s − 0.867·73-s + 1.86·77-s − 0.670·79-s + 0.499·83-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 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 - 0.597T + 13T^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
29 \( 1 + 4.64T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 - 1.80T + 47T^{2} \)
53 \( 1 + 8.13T + 53T^{2} \)
59 \( 1 - 3.80T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 - 0.0987T + 67T^{2} \)
71 \( 1 + 9.51T + 71T^{2} \)
73 \( 1 + 7.41T + 73T^{2} \)
79 \( 1 + 5.96T + 79T^{2} \)
83 \( 1 - 4.55T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.42341717976796211412444375657, −6.75588262074481373069771525049, −6.26518909303368029676537099081, −5.24328476992891702396008068409, −4.47795753960811261534009688652, −4.10240809599903416806967830271, −3.20301535088844592094083587369, −1.93266073219940448229579817567, −1.47657294554963078515004971132, 0, 1.47657294554963078515004971132, 1.93266073219940448229579817567, 3.20301535088844592094083587369, 4.10240809599903416806967830271, 4.47795753960811261534009688652, 5.24328476992891702396008068409, 6.26518909303368029676537099081, 6.75588262074481373069771525049, 7.42341717976796211412444375657

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