Properties

Label 2-588-1.1-c3-0-11
Degree $2$
Conductor $588$
Sign $-1$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 19.1·5-s + 9·9-s + 40.5·11-s + 50.4·13-s + 57.4·15-s − 51.9·17-s − 33.1·19-s + 62.8·23-s + 241.·25-s − 27·27-s + 129.·29-s + 242.·31-s − 121.·33-s − 389.·37-s − 151.·39-s − 470.·41-s − 125.·43-s − 172.·45-s − 386.·47-s + 155.·51-s − 611.·53-s − 777.·55-s + 99.3·57-s + 226.·59-s + 725.·61-s − 966.·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.71·5-s + 0.333·9-s + 1.11·11-s + 1.07·13-s + 0.988·15-s − 0.740·17-s − 0.400·19-s + 0.569·23-s + 1.93·25-s − 0.192·27-s + 0.831·29-s + 1.40·31-s − 0.642·33-s − 1.73·37-s − 0.621·39-s − 1.79·41-s − 0.443·43-s − 0.570·45-s − 1.19·47-s + 0.427·51-s − 1.58·53-s − 1.90·55-s + 0.230·57-s + 0.499·59-s + 1.52·61-s − 1.84·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 + 19.1T + 125T^{2} \)
11 \( 1 - 40.5T + 1.33e3T^{2} \)
13 \( 1 - 50.4T + 2.19e3T^{2} \)
17 \( 1 + 51.9T + 4.91e3T^{2} \)
19 \( 1 + 33.1T + 6.85e3T^{2} \)
23 \( 1 - 62.8T + 1.21e4T^{2} \)
29 \( 1 - 129.T + 2.43e4T^{2} \)
31 \( 1 - 242.T + 2.97e4T^{2} \)
37 \( 1 + 389.T + 5.06e4T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + 386.T + 1.03e5T^{2} \)
53 \( 1 + 611.T + 1.48e5T^{2} \)
59 \( 1 - 226.T + 2.05e5T^{2} \)
61 \( 1 - 725.T + 2.26e5T^{2} \)
67 \( 1 - 1.04e3T + 3.00e5T^{2} \)
71 \( 1 - 169.T + 3.57e5T^{2} \)
73 \( 1 + 381.T + 3.89e5T^{2} \)
79 \( 1 + 1.16e3T + 4.93e5T^{2} \)
83 \( 1 + 808.T + 5.71e5T^{2} \)
89 \( 1 + 319.T + 7.04e5T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−10.01085102774548040088387791752, −8.575806600707309894970055250865, −8.349575065348513017202112003208, −6.85099992104691433400277791690, −6.58998219620040408884994300913, −4.97825154854960192309880990932, −4.11723246662222883427579403099, −3.33342757741722071905706309588, −1.27372297138780187910107890154, 0, 1.27372297138780187910107890154, 3.33342757741722071905706309588, 4.11723246662222883427579403099, 4.97825154854960192309880990932, 6.58998219620040408884994300913, 6.85099992104691433400277791690, 8.349575065348513017202112003208, 8.575806600707309894970055250865, 10.01085102774548040088387791752

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