Properties

Label 2-4608-8.5-c1-0-35
Degree $2$
Conductor $4608$
Sign $1$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·5-s + 1.41·7-s + 2i·11-s − 2·17-s + 4i·19-s − 2.82·23-s + 2.99·25-s − 9.89i·29-s + 7.07·31-s − 2.00i·35-s + 8.48i·37-s + 6·41-s − 8i·43-s + 2.82·47-s − 5·49-s + ⋯
L(s)  = 1  − 0.632i·5-s + 0.534·7-s + 0.603i·11-s − 0.485·17-s + 0.917i·19-s − 0.589·23-s + 0.599·25-s − 1.83i·29-s + 1.27·31-s − 0.338i·35-s + 1.39i·37-s + 0.937·41-s − 1.21i·43-s + 0.412·47-s − 0.714·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013607158\)
\(L(\frac12)\) \(\approx\) \(2.013607158\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 9.89iT - 29T^{2} \)
31 \( 1 - 7.07T + 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 1.41iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 14.1iT - 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.268535267744260627670584311593, −7.78423797141011679171442599603, −6.87583827286314315173794216136, −6.08474469930001689170614329683, −5.34979043634966811847091014094, −4.44413664508296496714480593828, −4.11395722423283056664546112277, −2.75109823258849035998954484800, −1.88860240265936070572036459358, −0.854833749996929003678924908984, 0.75062795931874660358265460023, 2.03067494994519599869099220131, 2.89722773300279085305839051145, 3.67143712359787460018648099296, 4.71644448163048735560586080506, 5.25175197735978266990159926893, 6.40880769833428231566551980036, 6.67826571613589883126997453534, 7.68472034845463256759112926822, 8.221894519699595911132818334118

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