Properties

Label 2-588-1.1-c5-0-11
Degree $2$
Conductor $588$
Sign $1$
Analytic cond. $94.3056$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 6·5-s + 81·9-s − 108·11-s + 346·13-s − 54·15-s + 1.39e3·17-s + 1.01e3·19-s − 1.53e3·23-s − 3.08e3·25-s + 729·27-s − 3.76e3·29-s + 736·31-s − 972·33-s + 2.05e3·37-s + 3.11e3·39-s + 1.55e4·41-s + 1.10e4·43-s − 486·45-s − 4.56e3·47-s + 1.25e4·51-s − 7.96e3·53-s + 648·55-s + 9.10e3·57-s + 7.02e3·59-s − 2.68e4·61-s − 2.07e3·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.107·5-s + 1/3·9-s − 0.269·11-s + 0.567·13-s − 0.0619·15-s + 1.17·17-s + 0.643·19-s − 0.605·23-s − 0.988·25-s + 0.192·27-s − 0.830·29-s + 0.137·31-s − 0.155·33-s + 0.246·37-s + 0.327·39-s + 1.44·41-s + 0.910·43-s − 0.0357·45-s − 0.301·47-s + 0.677·51-s − 0.389·53-s + 0.0288·55-s + 0.371·57-s + 0.262·59-s − 0.924·61-s − 0.0609·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.815282936\)
\(L(\frac12)\) \(\approx\) \(2.815282936\)
\(L(\frac{7}{2})\) 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 \( 1 - p^{2} T \)
7 \( 1 \)
good5 \( 1 + 6 T + p^{5} T^{2} \)
11 \( 1 + 108 T + p^{5} T^{2} \)
13 \( 1 - 346 T + p^{5} T^{2} \)
17 \( 1 - 1398 T + p^{5} T^{2} \)
19 \( 1 - 1012 T + p^{5} T^{2} \)
23 \( 1 + 1536 T + p^{5} T^{2} \)
29 \( 1 + 3762 T + p^{5} T^{2} \)
31 \( 1 - 736 T + p^{5} T^{2} \)
37 \( 1 - 2054 T + p^{5} T^{2} \)
41 \( 1 - 15534 T + p^{5} T^{2} \)
43 \( 1 - 11036 T + p^{5} T^{2} \)
47 \( 1 + 4560 T + p^{5} T^{2} \)
53 \( 1 + 7962 T + p^{5} T^{2} \)
59 \( 1 - 7020 T + p^{5} T^{2} \)
61 \( 1 + 26870 T + p^{5} T^{2} \)
67 \( 1 - 52148 T + p^{5} T^{2} \)
71 \( 1 + 2544 T + p^{5} T^{2} \)
73 \( 1 - 9766 T + p^{5} T^{2} \)
79 \( 1 - 68672 T + p^{5} T^{2} \)
83 \( 1 - 61668 T + p^{5} T^{2} \)
89 \( 1 - 41454 T + p^{5} T^{2} \)
97 \( 1 - 111262 T + p^{5} T^{2} \)
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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

−9.773083523589339363394772116270, −9.137834853713180035317126987273, −7.915915389194582265773586661602, −7.62738775552654160588568924937, −6.24438827215191373990848947083, −5.37379250372251904639380530811, −4.07002404696707268652485412090, −3.25409774896807186593900294124, −2.04397226402739159979435903329, −0.809272566730893268332528956863, 0.809272566730893268332528956863, 2.04397226402739159979435903329, 3.25409774896807186593900294124, 4.07002404696707268652485412090, 5.37379250372251904639380530811, 6.24438827215191373990848947083, 7.62738775552654160588568924937, 7.915915389194582265773586661602, 9.137834853713180035317126987273, 9.773083523589339363394772116270

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