Properties

Label 2-4608-1.1-c1-0-2
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  − 1.41·5-s − 3.16·7-s − 4.47·11-s − 4.47·13-s − 6.32·17-s − 2.82·19-s + 4·23-s − 2.99·25-s + 4.24·29-s + 3.16·31-s + 4.47·35-s − 4.47·37-s − 6.32·41-s − 8.48·43-s + 12·47-s + 3.00·49-s − 7.07·53-s + 6.32·55-s + 13.4·61-s + 6.32·65-s + 8·71-s − 4·73-s + 14.1·77-s − 3.16·79-s + 4.47·83-s + 8.94·85-s + 14.1·91-s + ⋯
L(s)  = 1  − 0.632·5-s − 1.19·7-s − 1.34·11-s − 1.24·13-s − 1.53·17-s − 0.648·19-s + 0.834·23-s − 0.599·25-s + 0.787·29-s + 0.567·31-s + 0.755·35-s − 0.735·37-s − 0.987·41-s − 1.29·43-s + 1.75·47-s + 0.428·49-s − 0.971·53-s + 0.852·55-s + 1.71·61-s + 0.784·65-s + 0.949·71-s − 0.468·73-s + 1.61·77-s − 0.355·79-s + 0.490·83-s + 0.970·85-s + 1.48·91-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

\(L(1)\) \(\approx\) \(0.3356443789\)
\(L(\frac12)\) \(\approx\) \(0.3356443789\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 6.32T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 3.16T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 6.32T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 3.16T + 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.359095041027263743923829806799, −7.51396789235878965645859329907, −6.87828680084021580102321179316, −6.34242268729868722537302321711, −5.20653465278591567684036338619, −4.67771885008315949356852498337, −3.72560343477229900593509817582, −2.82204764569431329316125914538, −2.23051793818484345219107273088, −0.29442274249781570382213466106, 0.29442274249781570382213466106, 2.23051793818484345219107273088, 2.82204764569431329316125914538, 3.72560343477229900593509817582, 4.67771885008315949356852498337, 5.20653465278591567684036338619, 6.34242268729868722537302321711, 6.87828680084021580102321179316, 7.51396789235878965645859329907, 8.359095041027263743923829806799

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