Properties

Label 2-6048-1.1-c1-0-13
Degree $2$
Conductor $6048$
Sign $1$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
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  − 2.29·5-s − 7-s + 5.01·11-s − 4.55·13-s − 0.762·17-s + 3.01·19-s − 1.53·23-s + 0.278·25-s − 3.29·29-s − 3.29·31-s + 2.29·35-s + 4.72·37-s − 0.297·41-s + 3.83·43-s + 5.79·47-s + 49-s − 11.3·53-s − 11.5·55-s − 3.59·59-s + 2.42·61-s + 10.4·65-s − 4.12·67-s − 2.46·71-s + 16.6·73-s − 5.01·77-s + 5.36·79-s + 1.27·83-s + ⋯
L(s)  = 1  − 1.02·5-s − 0.377·7-s + 1.51·11-s − 1.26·13-s − 0.185·17-s + 0.692·19-s − 0.319·23-s + 0.0557·25-s − 0.612·29-s − 0.592·31-s + 0.388·35-s + 0.776·37-s − 0.0464·41-s + 0.584·43-s + 0.844·47-s + 0.142·49-s − 1.56·53-s − 1.55·55-s − 0.468·59-s + 0.310·61-s + 1.29·65-s − 0.504·67-s − 0.292·71-s + 1.94·73-s − 0.571·77-s + 0.603·79-s + 0.140·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210402451\)
\(L(\frac12)\) \(\approx\) \(1.210402451\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 2.29T + 5T^{2} \)
11 \( 1 - 5.01T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 + 0.762T + 17T^{2} \)
19 \( 1 - 3.01T + 19T^{2} \)
23 \( 1 + 1.53T + 23T^{2} \)
29 \( 1 + 3.29T + 29T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 + 0.297T + 41T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 - 5.79T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 3.59T + 59T^{2} \)
61 \( 1 - 2.42T + 61T^{2} \)
67 \( 1 + 4.12T + 67T^{2} \)
71 \( 1 + 2.46T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 - 5.36T + 79T^{2} \)
83 \( 1 - 1.27T + 83T^{2} \)
89 \( 1 + 9.57T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−7.83837043086821579816637181219, −7.47079102265816505308068774998, −6.75758365610358618864630042073, −6.04811687320886628760728259605, −5.11576257541544519760119368618, −4.25080306713208167534414791718, −3.77246648103971959206995557915, −2.92396102183980194736626359892, −1.82059380546967648439422623439, −0.57199795422711043454293986921, 0.57199795422711043454293986921, 1.82059380546967648439422623439, 2.92396102183980194736626359892, 3.77246648103971959206995557915, 4.25080306713208167534414791718, 5.11576257541544519760119368618, 6.04811687320886628760728259605, 6.75758365610358618864630042073, 7.47079102265816505308068774998, 7.83837043086821579816637181219

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