Properties

Label 2-6048-1.1-c1-0-60
Degree $2$
Conductor $6048$
Sign $-1$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s + 11-s + 2·13-s − 2·17-s − 3·19-s + 23-s + 4·25-s + 2·29-s − 5·31-s − 3·35-s + 7·37-s − 7·41-s + 8·43-s − 4·47-s + 49-s + 4·53-s − 3·55-s − 4·59-s + 4·61-s − 6·65-s + 2·67-s − 3·71-s + 77-s + 4·79-s − 2·83-s + 6·85-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.688·19-s + 0.208·23-s + 4/5·25-s + 0.371·29-s − 0.898·31-s − 0.507·35-s + 1.15·37-s − 1.09·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.549·53-s − 0.404·55-s − 0.520·59-s + 0.512·61-s − 0.744·65-s + 0.244·67-s − 0.356·71-s + 0.113·77-s + 0.450·79-s − 0.219·83-s + 0.650·85-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−7.78647946075266095106869546071, −7.09040679489009066280908180885, −6.43023330892861486057843263388, −5.55404355303695336094919181167, −4.58892478666671379745899471265, −4.08828945983619711306818541888, −3.40801457741173171560993204111, −2.36509861745557567121323901143, −1.20070144905169503309757912188, 0, 1.20070144905169503309757912188, 2.36509861745557567121323901143, 3.40801457741173171560993204111, 4.08828945983619711306818541888, 4.58892478666671379745899471265, 5.55404355303695336094919181167, 6.43023330892861486057843263388, 7.09040679489009066280908180885, 7.78647946075266095106869546071

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