Properties

Label 2-588-1.1-c3-0-15
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 + 10.6·5-s + 9·9-s − 6.65·11-s − 75.9·13-s − 31.9·15-s + 104.·17-s − 85.4·19-s − 68.6·23-s − 11.4·25-s − 27·27-s + 87.7·29-s − 62.7·31-s + 19.9·33-s + 42.2·37-s + 227.·39-s − 313.·41-s + 306.·43-s + 95.8·45-s − 215.·47-s − 312.·51-s + 525.·53-s − 70.8·55-s + 256.·57-s − 360.·59-s − 800.·61-s − 809.·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.953·5-s + 0.333·9-s − 0.182·11-s − 1.62·13-s − 0.550·15-s + 1.48·17-s − 1.03·19-s − 0.622·23-s − 0.0917·25-s − 0.192·27-s + 0.562·29-s − 0.363·31-s + 0.105·33-s + 0.187·37-s + 0.935·39-s − 1.19·41-s + 1.08·43-s + 0.317·45-s − 0.667·47-s − 0.859·51-s + 1.36·53-s − 0.173·55-s + 0.595·57-s − 0.795·59-s − 1.68·61-s − 1.54·65-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 \( 1 - 10.6T + 125T^{2} \)
11 \( 1 + 6.65T + 1.33e3T^{2} \)
13 \( 1 + 75.9T + 2.19e3T^{2} \)
17 \( 1 - 104.T + 4.91e3T^{2} \)
19 \( 1 + 85.4T + 6.85e3T^{2} \)
23 \( 1 + 68.6T + 1.21e4T^{2} \)
29 \( 1 - 87.7T + 2.43e4T^{2} \)
31 \( 1 + 62.7T + 2.97e4T^{2} \)
37 \( 1 - 42.2T + 5.06e4T^{2} \)
41 \( 1 + 313.T + 6.89e4T^{2} \)
43 \( 1 - 306.T + 7.95e4T^{2} \)
47 \( 1 + 215.T + 1.03e5T^{2} \)
53 \( 1 - 525.T + 1.48e5T^{2} \)
59 \( 1 + 360.T + 2.05e5T^{2} \)
61 \( 1 + 800.T + 2.26e5T^{2} \)
67 \( 1 + 40.2T + 3.00e5T^{2} \)
71 \( 1 + 298.T + 3.57e5T^{2} \)
73 \( 1 + 517.T + 3.89e5T^{2} \)
79 \( 1 + 1.22e3T + 4.93e5T^{2} \)
83 \( 1 + 1.32e3T + 5.71e5T^{2} \)
89 \( 1 - 639.T + 7.04e5T^{2} \)
97 \( 1 + 1.42e3T + 9.12e5T^{2} \)
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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

−10.12714049391522965508650122502, −9.209285164303310246279296468776, −7.960258571737601823551325484289, −7.09350741436190570309337792608, −6.02142619955825502294064851900, −5.37026543394511362096076561078, −4.35988837380580326433221385717, −2.77817681564899282027104408059, −1.63549290811764027845480793228, 0, 1.63549290811764027845480793228, 2.77817681564899282027104408059, 4.35988837380580326433221385717, 5.37026543394511362096076561078, 6.02142619955825502294064851900, 7.09350741436190570309337792608, 7.960258571737601823551325484289, 9.209285164303310246279296468776, 10.12714049391522965508650122502

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