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Properties

Label 2-78e2-1.1-c1-0-36
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  − 4·5-s + 2·7-s − 4·11-s − 2·17-s + 2·19-s + 11·25-s + 6·29-s + 10·31-s − 8·35-s − 10·37-s + 8·41-s + 4·43-s − 4·47-s − 3·49-s + 10·53-s + 16·55-s − 8·59-s − 14·61-s − 2·67-s + 16·71-s + 10·73-s − 8·77-s − 16·79-s + 8·85-s − 4·89-s − 8·95-s + 2·97-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.755·7-s − 1.20·11-s − 0.485·17-s + 0.458·19-s + 11/5·25-s + 1.11·29-s + 1.79·31-s − 1.35·35-s − 1.64·37-s + 1.24·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s + 2.15·55-s − 1.04·59-s − 1.79·61-s − 0.244·67-s + 1.89·71-s + 1.17·73-s − 0.911·77-s − 1.80·79-s + 0.867·85-s − 0.423·89-s − 0.820·95-s + 0.203·97-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 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−7.84976557281064469235677567853, −7.22285912063297431682129960074, −6.46816364752302142158565755894, −5.32117630855474907985926765020, −4.69595250185142144057975790869, −4.19998611240233753992623256824, −3.20275390410942049767079678171, −2.53883903349824836372556939282, −1.09634080412413673261083505900, 0, 1.09634080412413673261083505900, 2.53883903349824836372556939282, 3.20275390410942049767079678171, 4.19998611240233753992623256824, 4.69595250185142144057975790869, 5.32117630855474907985926765020, 6.46816364752302142158565755894, 7.22285912063297431682129960074, 7.84976557281064469235677567853

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