Properties

Label 2-66e2-1.1-c1-0-13
Degree $2$
Conductor $4356$
Sign $1$
Analytic cond. $34.7828$
Root an. cond. $5.89769$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·7-s + 2·13-s + 4·17-s + 6·19-s − 25-s − 8·29-s − 8·31-s − 4·35-s + 10·37-s + 8·41-s + 2·43-s + 8·47-s − 3·49-s + 2·53-s − 12·59-s − 10·61-s − 4·65-s + 12·67-s − 8·71-s − 6·73-s + 2·79-s + 16·83-s − 8·85-s + 14·89-s + 4·91-s − 12·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 1/5·25-s − 1.48·29-s − 1.43·31-s − 0.676·35-s + 1.64·37-s + 1.24·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.225·79-s + 1.75·83-s − 0.867·85-s + 1.48·89-s + 0.419·91-s − 1.23·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(34.7828\)
Root analytic conductor: \(5.89769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4356,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.849630212\)
\(L(\frac12)\) \(\approx\) \(1.849630212\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−8.074379069264153068708952280908, −7.66700790333795020456742124482, −7.30223796795303190064148405091, −5.95818902421944129428630267000, −5.51331429337006714198744949212, −4.52702091907548133307727064364, −3.79955930348242148717135072932, −3.11474424285935633940176598416, −1.82500088072096369066062448326, −0.796015146435045811880856861272, 0.796015146435045811880856861272, 1.82500088072096369066062448326, 3.11474424285935633940176598416, 3.79955930348242148717135072932, 4.52702091907548133307727064364, 5.51331429337006714198744949212, 5.95818902421944129428630267000, 7.30223796795303190064148405091, 7.66700790333795020456742124482, 8.074379069264153068708952280908

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