Properties

Label 2-4608-1.1-c1-0-39
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  + 3.41·5-s + 0.585·7-s + 2·11-s + 2.82·13-s + 7.65·17-s − 5.65·19-s − 6.82·23-s + 6.65·25-s + 3.41·29-s + 7.41·31-s + 2·35-s − 1.65·37-s + 0.343·41-s + 9.65·43-s + 4.48·47-s − 6.65·49-s − 7.89·53-s + 6.82·55-s − 4·59-s − 1.65·61-s + 9.65·65-s + 8·67-s − 14.8·71-s + 9.65·73-s + 1.17·77-s + 14.2·79-s − 13.3·83-s + ⋯
L(s)  = 1  + 1.52·5-s + 0.221·7-s + 0.603·11-s + 0.784·13-s + 1.85·17-s − 1.29·19-s − 1.42·23-s + 1.33·25-s + 0.634·29-s + 1.33·31-s + 0.338·35-s − 0.272·37-s + 0.0535·41-s + 1.47·43-s + 0.654·47-s − 0.950·49-s − 1.08·53-s + 0.920·55-s − 0.520·59-s − 0.212·61-s + 1.19·65-s + 0.977·67-s − 1.75·71-s + 1.13·73-s + 0.133·77-s + 1.60·79-s − 1.46·83-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

\(L(1)\) \(\approx\) \(3.151505303\)
\(L(\frac12)\) \(\approx\) \(3.151505303\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 - 0.585T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 - 3.41T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 - 0.343T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 + 7.89T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 9.65T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 9.31T + 97T^{2} \)
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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

−8.331855736467897710675438440363, −7.73767693179721289177421371333, −6.51938349739942481526667437844, −6.15238525488398953993879243620, −5.61020677836677848325583594990, −4.64301621794781698873241075286, −3.78443940346819520216077468930, −2.75837890263118224056386816817, −1.83619650703641114579002383234, −1.08709898077617771084790018006, 1.08709898077617771084790018006, 1.83619650703641114579002383234, 2.75837890263118224056386816817, 3.78443940346819520216077468930, 4.64301621794781698873241075286, 5.61020677836677848325583594990, 6.15238525488398953993879243620, 6.51938349739942481526667437844, 7.73767693179721289177421371333, 8.331855736467897710675438440363

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