Properties

Label 2-4608-1.1-c1-0-25
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 yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.585·5-s + 3.41·7-s + 2·11-s − 2.82·13-s − 3.65·17-s + 5.65·19-s − 1.17·23-s − 4.65·25-s + 0.585·29-s + 4.58·31-s + 2·35-s + 9.65·37-s + 11.6·41-s − 1.65·43-s − 12.4·47-s + 4.65·49-s + 11.8·53-s + 1.17·55-s − 4·59-s + 9.65·61-s − 1.65·65-s + 8·67-s − 9.17·71-s − 1.65·73-s + 6.82·77-s + 5.75·79-s + 9.31·83-s + ⋯
L(s)  = 1  + 0.261·5-s + 1.29·7-s + 0.603·11-s − 0.784·13-s − 0.886·17-s + 1.29·19-s − 0.244·23-s − 0.931·25-s + 0.108·29-s + 0.823·31-s + 0.338·35-s + 1.58·37-s + 1.82·41-s − 0.252·43-s − 1.82·47-s + 0.665·49-s + 1.63·53-s + 0.157·55-s − 0.520·59-s + 1.23·61-s − 0.205·65-s + 0.977·67-s − 1.08·71-s − 0.193·73-s + 0.778·77-s + 0.647·79-s + 1.02·83-s + ⋯

Functional equation

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

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: $\chi_{4608} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.498079988\)
\(L(\frac12)\) \(\approx\) \(2.498079988\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585T + 5T^{2} \)
7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 - 0.585T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 9.17T + 71T^{2} \)
73 \( 1 + 1.65T + 73T^{2} \)
79 \( 1 - 5.75T + 79T^{2} \)
83 \( 1 - 9.31T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−8.115986525583857272656340209959, −7.76286562195232263396088706111, −6.91975511187405688868801555274, −6.10361659804626254678279783841, −5.30034851041620303420082673160, −4.61692563488086584660718316133, −3.96683080443626455630115761002, −2.70841013164296153844322149236, −1.93723054310808572352251286324, −0.918619376833668659725951051311, 0.918619376833668659725951051311, 1.93723054310808572352251286324, 2.70841013164296153844322149236, 3.96683080443626455630115761002, 4.61692563488086584660718316133, 5.30034851041620303420082673160, 6.10361659804626254678279783841, 6.91975511187405688868801555274, 7.76286562195232263396088706111, 8.115986525583857272656340209959

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