Properties

Label 2-8280-1.1-c1-0-1
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 − 2.07·7-s − 5.07·11-s − 3.48·13-s − 6.48·17-s − 7.48·19-s − 23-s + 25-s + 1.56·29-s + 0.0791·31-s + 2.07·35-s − 9.71·37-s + 0.480·41-s + 8·43-s − 6.96·47-s − 2.67·49-s − 11.7·53-s + 5.07·55-s + 11.5·59-s + 7.88·61-s + 3.48·65-s − 9.71·67-s + 9.67·71-s − 13.2·73-s + 10.5·77-s − 12.3·79-s + 4.59·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.785·7-s − 1.53·11-s − 0.965·13-s − 1.57·17-s − 1.71·19-s − 0.208·23-s + 0.200·25-s + 0.289·29-s + 0.0142·31-s + 0.351·35-s − 1.59·37-s + 0.0751·41-s + 1.21·43-s − 1.01·47-s − 0.382·49-s − 1.60·53-s + 0.684·55-s + 1.50·59-s + 1.00·61-s + 0.431·65-s − 1.18·67-s + 1.14·71-s − 1.55·73-s + 1.20·77-s − 1.38·79-s + 0.504·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1083942512\)
\(L(\frac12)\) \(\approx\) \(0.1083942512\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.07T + 7T^{2} \)
11 \( 1 + 5.07T + 11T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 + 6.48T + 17T^{2} \)
19 \( 1 + 7.48T + 19T^{2} \)
29 \( 1 - 1.56T + 29T^{2} \)
31 \( 1 - 0.0791T + 31T^{2} \)
37 \( 1 + 9.71T + 37T^{2} \)
41 \( 1 - 0.480T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 + 9.71T + 67T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 - 7.23T + 97T^{2} \)
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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

−7.79634044404134290654461363384, −7.05863636150598934177994427581, −6.55913264205417122304155305226, −5.77672117297107931255997652604, −4.83000807713066451680847828344, −4.43618959967067953824648731457, −3.43368101085530767956384071677, −2.58546679564020879664825248617, −2.04832530416573198215262039364, −0.14697329336062861772474524143, 0.14697329336062861772474524143, 2.04832530416573198215262039364, 2.58546679564020879664825248617, 3.43368101085530767956384071677, 4.43618959967067953824648731457, 4.83000807713066451680847828344, 5.77672117297107931255997652604, 6.55913264205417122304155305226, 7.05863636150598934177994427581, 7.79634044404134290654461363384

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