Properties

Label 2-78e2-1.1-c1-0-62
Degree $2$
Conductor $6084$
Sign $-1$
Analytic cond. $48.5809$
Root an. cond. $6.97000$
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  + 2·5-s + 2·7-s − 2·11-s − 6·17-s + 6·19-s − 8·23-s − 25-s − 2·29-s − 10·31-s + 4·35-s + 6·37-s − 6·41-s + 4·43-s − 2·47-s − 3·49-s − 6·53-s − 4·55-s − 10·59-s − 2·61-s − 10·67-s + 10·71-s − 2·73-s − 4·77-s − 4·79-s − 6·83-s − 12·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s − 0.603·11-s − 1.45·17-s + 1.37·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s − 1.79·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s − 1.30·59-s − 0.256·61-s − 1.22·67-s + 1.18·71-s − 0.234·73-s − 0.455·77-s − 0.450·79-s − 0.658·83-s − 1.30·85-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6084\)    =    \(2^{2} \cdot 3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(48.5809\)
Root analytic conductor: \(6.97000\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6084,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.75278361549181880377727071254, −7.08998532903080357642556758525, −6.10014918731261564995074453282, −5.64953725554715039802248200782, −4.89233148520829001479133968847, −4.16337264345952282915929548204, −3.12141642932320178028594426715, −2.09132246047893191349158801492, −1.63392338428425511982320095775, 0, 1.63392338428425511982320095775, 2.09132246047893191349158801492, 3.12141642932320178028594426715, 4.16337264345952282915929548204, 4.89233148520829001479133968847, 5.64953725554715039802248200782, 6.10014918731261564995074453282, 7.08998532903080357642556758525, 7.75278361549181880377727071254

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