Properties

Label 2-52e2-1.1-c1-0-21
Degree $2$
Conductor $2704$
Sign $1$
Analytic cond. $21.5915$
Root an. cond. $4.64667$
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  − 3-s + 3·5-s − 7-s − 2·9-s + 6·11-s − 3·15-s − 3·17-s + 2·19-s + 21-s + 4·25-s + 5·27-s + 6·29-s − 4·31-s − 6·33-s − 3·35-s + 7·37-s + 43-s − 6·45-s + 3·47-s − 6·49-s + 3·51-s + 18·55-s − 2·57-s − 6·59-s + 8·61-s + 2·63-s + 14·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 0.377·7-s − 2/3·9-s + 1.80·11-s − 0.774·15-s − 0.727·17-s + 0.458·19-s + 0.218·21-s + 4/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.507·35-s + 1.15·37-s + 0.152·43-s − 0.894·45-s + 0.437·47-s − 6/7·49-s + 0.420·51-s + 2.42·55-s − 0.264·57-s − 0.781·59-s + 1.02·61-s + 0.251·63-s + 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2704\)    =    \(2^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.5915\)
Root analytic conductor: \(4.64667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2704,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.929992512\)
\(L(\frac12)\) \(\approx\) \(1.929992512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.151174717145742030269782508872, −8.244134846950840981107676678567, −6.92421875375835189696702665586, −6.40028526578024760134946435749, −5.93814316414134510481853694029, −5.11332595609013512036605695413, −4.14534392918182308978597121442, −3.04804615759780815838301814050, −2.01914482841960457865874156726, −0.925636388633133141442280821607, 0.925636388633133141442280821607, 2.01914482841960457865874156726, 3.04804615759780815838301814050, 4.14534392918182308978597121442, 5.11332595609013512036605695413, 5.93814316414134510481853694029, 6.40028526578024760134946435749, 6.92421875375835189696702665586, 8.244134846950840981107676678567, 9.151174717145742030269782508872

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