Properties

Label 2-4608-1.1-c1-0-36
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  + 2.08·5-s + 5.03·7-s − 0.828·11-s + 2.94·13-s − 4.82·17-s − 2.82·19-s + 4.16·23-s − 0.656·25-s + 7.97·29-s − 5.03·31-s + 10.4·35-s + 7.11·37-s − 8.82·41-s + 12.4·43-s + 4.16·47-s + 18.3·49-s − 12.1·53-s − 1.72·55-s − 1.65·59-s + 7.11·61-s + 6.14·65-s + 2.34·67-s + 10.0·71-s + 4·73-s − 4.16·77-s + 5.03·79-s + 3.17·83-s + ⋯
L(s)  = 1  + 0.932·5-s + 1.90·7-s − 0.249·11-s + 0.817·13-s − 1.17·17-s − 0.648·19-s + 0.869·23-s − 0.131·25-s + 1.48·29-s − 0.903·31-s + 1.77·35-s + 1.16·37-s − 1.37·41-s + 1.90·43-s + 0.607·47-s + 2.61·49-s − 1.66·53-s − 0.232·55-s − 0.215·59-s + 0.911·61-s + 0.761·65-s + 0.286·67-s + 1.19·71-s + 0.468·73-s − 0.474·77-s + 0.566·79-s + 0.348·83-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

\(L(1)\) \(\approx\) \(3.115022678\)
\(L(\frac12)\) \(\approx\) \(3.115022678\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.08T + 5T^{2} \)
7 \( 1 - 5.03T + 7T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 - 2.94T + 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 4.16T + 23T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 + 5.03T + 31T^{2} \)
37 \( 1 - 7.11T + 37T^{2} \)
41 \( 1 + 8.82T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 - 7.11T + 61T^{2} \)
67 \( 1 - 2.34T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−8.368710710601918342097557266308, −7.74993355118010709600460929060, −6.79135401542428140374401381131, −6.10486751256755172073102709834, −5.29447791588718935174009823266, −4.70627976007212790487739223705, −3.98700225523211816828518473187, −2.57002486619137647485707925793, −1.93600191184547583681500063077, −1.06480502007850324764970031627, 1.06480502007850324764970031627, 1.93600191184547583681500063077, 2.57002486619137647485707925793, 3.98700225523211816828518473187, 4.70627976007212790487739223705, 5.29447791588718935174009823266, 6.10486751256755172073102709834, 6.79135401542428140374401381131, 7.74993355118010709600460929060, 8.368710710601918342097557266308

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