Properties

Label 2-4608-1.1-c1-0-19
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  − 3.95·5-s + 1.63·7-s + 4.82·11-s + 5.59·13-s + 0.828·17-s + 2.82·19-s − 7.91·23-s + 10.6·25-s + 7.23·29-s − 1.63·31-s − 6.48·35-s − 2.31·37-s − 3.17·41-s − 4.48·43-s − 7.91·47-s − 4.31·49-s + 0.678·53-s − 19.1·55-s + 9.65·59-s − 2.31·61-s − 22.1·65-s + 13.6·67-s + 3.27·71-s + 4·73-s + 7.91·77-s + 1.63·79-s + 8.82·83-s + ⋯
L(s)  = 1  − 1.76·5-s + 0.619·7-s + 1.45·11-s + 1.55·13-s + 0.200·17-s + 0.648·19-s − 1.65·23-s + 2.13·25-s + 1.34·29-s − 0.294·31-s − 1.09·35-s − 0.381·37-s − 0.495·41-s − 0.683·43-s − 1.15·47-s − 0.616·49-s + 0.0932·53-s − 2.57·55-s + 1.25·59-s − 0.296·61-s − 2.74·65-s + 1.66·67-s + 0.389·71-s + 0.468·73-s + 0.901·77-s + 0.184·79-s + 0.969·83-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

\(L(1)\) \(\approx\) \(1.713881640\)
\(L(\frac12)\) \(\approx\) \(1.713881640\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.95T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 7.91T + 23T^{2} \)
29 \( 1 - 7.23T + 29T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 + 3.17T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + 7.91T + 47T^{2} \)
53 \( 1 - 0.678T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 1.63T + 79T^{2} \)
83 \( 1 - 8.82T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.255945157988399077416176495398, −7.84862527322874588103678532867, −6.79883359360601630486828198693, −6.39817046818188230719409036889, −5.24580544107918271994435847522, −4.35276585406584148815087713272, −3.76412311754615987972267465554, −3.32731724529690228650556818709, −1.68524460678040110668288651810, −0.78036567888018742138106023790, 0.78036567888018742138106023790, 1.68524460678040110668288651810, 3.32731724529690228650556818709, 3.76412311754615987972267465554, 4.35276585406584148815087713272, 5.24580544107918271994435847522, 6.39817046818188230719409036889, 6.79883359360601630486828198693, 7.84862527322874588103678532867, 8.255945157988399077416176495398

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