Properties

Label 2-588-7.4-c5-0-23
Degree $2$
Conductor $588$
Sign $0.266 + 0.963i$
Analytic cond. $94.3056$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 7.79i)3-s + (−17 − 29.4i)5-s + (−40.5 − 70.1i)9-s + (166 − 287. i)11-s + 1.02e3·13-s − 306·15-s + (461 − 798. i)17-s + (226 + 391. i)19-s + (1.88e3 + 3.27e3i)23-s + (984.5 − 1.70e3i)25-s − 729·27-s + 1.16e3·29-s + (−4.89e3 + 8.48e3i)31-s + (−1.49e3 − 2.58e3i)33-s + (−1.19e3 − 2.06e3i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.304 − 0.526i)5-s + (−0.166 − 0.288i)9-s + (0.413 − 0.716i)11-s + 1.68·13-s − 0.351·15-s + (0.386 − 0.670i)17-s + (0.143 + 0.248i)19-s + (0.744 + 1.28i)23-s + (0.315 − 0.545i)25-s − 0.192·27-s + 0.257·29-s + (−0.915 + 1.58i)31-s + (−0.238 − 0.413i)33-s + (−0.143 − 0.248i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.741702564\)
\(L(\frac12)\) \(\approx\) \(2.741702564\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (17 + 29.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-166 + 287. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 1.02e3T + 3.71e5T^{2} \)
17 \( 1 + (-461 + 798. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-226 - 391. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.88e3 - 3.27e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 1.16e3T + 2.05e7T^{2} \)
31 \( 1 + (4.89e3 - 8.48e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (1.19e3 + 2.06e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 7.23e3T + 1.15e8T^{2} \)
43 \( 1 - 4.65e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.23e4 - 2.13e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (555 - 961. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.34e4 + 4.06e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.88e3 + 8.45e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.31e4 + 2.27e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.54e4T + 1.80e9T^{2} \)
73 \( 1 + (2.80e3 - 4.85e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.92e3 - 8.52e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.11e4T + 3.93e9T^{2} \)
89 \( 1 + (3.14e4 + 5.45e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.78e4T + 8.58e9T^{2} \)
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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

−9.434824891745122419984065294869, −8.778074570163057877507852833064, −8.096891387053096270548866263471, −7.10274141324486456222338863923, −6.12644637395723930757046707358, −5.22168825446719444274542194233, −3.84632088306085466294044263234, −3.09821921420994812021126176586, −1.44321178541878259114200827147, −0.75944980829321898402086288270, 0.995619255154840588672015310130, 2.38595868765742204590771505893, 3.60211630900326925667652027265, 4.20894398792025486511339071699, 5.54117874671356423701372230269, 6.53475247485694311099989551014, 7.43608506073343121737789303452, 8.498396727634016014894141175242, 9.126141365531566457855009907065, 10.20416110408918513718029922972

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