Properties

Label 2-4608-1.1-c1-0-3
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  − 2.44·5-s − 1.41·7-s − 3.46·11-s − 4.89·13-s + 4·17-s − 6.92·19-s − 5.65·23-s + 0.999·25-s − 2.44·29-s + 1.41·31-s + 3.46·35-s − 4.89·37-s + 4·41-s + 6.92·43-s + 5.65·47-s − 5·49-s + 7.34·53-s + 8.48·55-s + 13.8·59-s − 4.89·61-s + 11.9·65-s − 11.3·71-s − 4·73-s + 4.89·77-s − 7.07·79-s − 10.3·83-s − 9.79·85-s + ⋯
L(s)  = 1  − 1.09·5-s − 0.534·7-s − 1.04·11-s − 1.35·13-s + 0.970·17-s − 1.58·19-s − 1.17·23-s + 0.199·25-s − 0.454·29-s + 0.254·31-s + 0.585·35-s − 0.805·37-s + 0.624·41-s + 1.05·43-s + 0.825·47-s − 0.714·49-s + 1.00·53-s + 1.14·55-s + 1.80·59-s − 0.627·61-s + 1.48·65-s − 1.34·71-s − 0.468·73-s + 0.558·77-s − 0.795·79-s − 1.14·83-s − 1.06·85-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

\(L(1)\) \(\approx\) \(0.4722981658\)
\(L(\frac12)\) \(\approx\) \(0.4722981658\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 4.89T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.179241106968710109652229542484, −7.56968261334377706744850402481, −7.14664586315026700853661347026, −6.07701350004751985649604477034, −5.39964650025734038334357711465, −4.43220427854209321098822670179, −3.88075901163517833051891350768, −2.88199546820798744855326852443, −2.13011582929890769845634910224, −0.35521274087390964463913027938, 0.35521274087390964463913027938, 2.13011582929890769845634910224, 2.88199546820798744855326852443, 3.88075901163517833051891350768, 4.43220427854209321098822670179, 5.39964650025734038334357711465, 6.07701350004751985649604477034, 7.14664586315026700853661347026, 7.56968261334377706744850402481, 8.179241106968710109652229542484

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