Properties

Label 2-588-1.1-c3-0-19
Degree $2$
Conductor $588$
Sign $-1$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4·5-s + 9·9-s − 20·11-s + 4·13-s − 12·15-s − 24·17-s − 44·19-s + 72·23-s − 109·25-s + 27·27-s − 38·29-s − 184·31-s − 60·33-s − 30·37-s + 12·39-s + 216·41-s − 164·43-s − 36·45-s − 520·47-s − 72·51-s − 146·53-s + 80·55-s − 132·57-s − 460·59-s − 628·61-s − 16·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.357·5-s + 1/3·9-s − 0.548·11-s + 0.0853·13-s − 0.206·15-s − 0.342·17-s − 0.531·19-s + 0.652·23-s − 0.871·25-s + 0.192·27-s − 0.243·29-s − 1.06·31-s − 0.316·33-s − 0.133·37-s + 0.0492·39-s + 0.822·41-s − 0.581·43-s − 0.119·45-s − 1.61·47-s − 0.197·51-s − 0.378·53-s + 0.196·55-s − 0.306·57-s − 1.01·59-s − 1.31·61-s − 0.0305·65-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: yes
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 \( 1 - p T \)
7 \( 1 \)
good5 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 + 20 T + p^{3} T^{2} \)
13 \( 1 - 4 T + p^{3} T^{2} \)
17 \( 1 + 24 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 - 72 T + p^{3} T^{2} \)
29 \( 1 + 38 T + p^{3} T^{2} \)
31 \( 1 + 184 T + p^{3} T^{2} \)
37 \( 1 + 30 T + p^{3} T^{2} \)
41 \( 1 - 216 T + p^{3} T^{2} \)
43 \( 1 + 164 T + p^{3} T^{2} \)
47 \( 1 + 520 T + p^{3} T^{2} \)
53 \( 1 + 146 T + p^{3} T^{2} \)
59 \( 1 + 460 T + p^{3} T^{2} \)
61 \( 1 + 628 T + p^{3} T^{2} \)
67 \( 1 - 556 T + p^{3} T^{2} \)
71 \( 1 - 592 T + p^{3} T^{2} \)
73 \( 1 + 1024 T + p^{3} T^{2} \)
79 \( 1 + 104 T + p^{3} T^{2} \)
83 \( 1 - 324 T + p^{3} T^{2} \)
89 \( 1 + 896 T + p^{3} T^{2} \)
97 \( 1 - 920 T + p^{3} T^{2} \)
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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

−9.758146342599299349987807920748, −8.937854132955415755208166696597, −8.066601279655500870802386265595, −7.35413697247318852710235778072, −6.28748821323924787352665280094, −5.08419188852858136360338434160, −4.03197056309687359862470851700, −2.99510834693268327129981417637, −1.76701576654351682478215612959, 0, 1.76701576654351682478215612959, 2.99510834693268327129981417637, 4.03197056309687359862470851700, 5.08419188852858136360338434160, 6.28748821323924787352665280094, 7.35413697247318852710235778072, 8.066601279655500870802386265595, 8.937854132955415755208166696597, 9.758146342599299349987807920748

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