Properties

Label 2-588-7.4-c7-0-21
Degree $2$
Conductor $588$
Sign $0.991 - 0.126i$
Analytic cond. $183.682$
Root an. cond. $13.5529$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 − 23.3i)3-s + (115. + 199. i)5-s + (−364.5 − 631. i)9-s + (2.51e3 − 4.35e3i)11-s − 1.37e4·13-s + 6.22e3·15-s + (−1.62e4 + 2.81e4i)17-s + (−4.82e3 − 8.35e3i)19-s + (−1.60e4 − 2.77e4i)23-s + (1.25e4 − 2.16e4i)25-s − 1.96e4·27-s + 1.03e5·29-s + (−1.20e5 + 2.09e5i)31-s + (−6.78e4 − 1.17e5i)33-s + (6.35e4 + 1.10e5i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.412 + 0.713i)5-s + (−0.166 − 0.288i)9-s + (0.569 − 0.986i)11-s − 1.73·13-s + 0.475·15-s + (−0.801 + 1.38i)17-s + (−0.161 − 0.279i)19-s + (−0.274 − 0.475i)23-s + (0.160 − 0.277i)25-s − 0.192·27-s + 0.789·29-s + (−0.729 + 1.26i)31-s + (−0.328 − 0.569i)33-s + (0.206 + 0.357i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(183.682\)
Root analytic conductor: \(13.5529\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :7/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.199834600\)
\(L(\frac12)\) \(\approx\) \(2.199834600\)
\(L(\frac{9}{2})\) 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 \( 1 + (-13.5 + 23.3i)T \)
7 \( 1 \)
good5 \( 1 + (-115. - 199. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-2.51e3 + 4.35e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 1.37e4T + 6.27e7T^{2} \)
17 \( 1 + (1.62e4 - 2.81e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (4.82e3 + 8.35e3i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (1.60e4 + 2.77e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 1.03e5T + 1.72e10T^{2} \)
31 \( 1 + (1.20e5 - 2.09e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-6.35e4 - 1.10e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 6.07e5T + 1.94e11T^{2} \)
43 \( 1 - 4.43e5T + 2.71e11T^{2} \)
47 \( 1 + (3.45e5 + 5.99e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (3.36e5 - 5.82e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.29e6 + 2.23e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (7.67e5 + 1.33e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-2.10e6 + 3.64e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 1.51e6T + 9.09e12T^{2} \)
73 \( 1 + (2.73e6 - 4.74e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-3.37e6 - 5.85e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 8.36e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.58e6 - 4.47e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 1.06e7T + 8.07e13T^{2} \)
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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

−9.548693622563398857202008216154, −8.660526105716530372933959782188, −7.83914599983247831827242876623, −6.67819930958281110405900214635, −6.36880760082286085766352469447, −5.06606387721668865347738731028, −3.83854065961228987615664497863, −2.71316836514523747267910325057, −2.02218711514296363753556364014, −0.67182538685986134988338598154, 0.54101375141594032539652796967, 1.94202140846319820868600808138, 2.70945315116140436374467654155, 4.28245979857490005993688445733, 4.75947312355415387385848733855, 5.72531207267670655034006966204, 7.08499570221674701479767063378, 7.69036139895041911715911477250, 9.113478915839894445553929972966, 9.405114054501019449195853129160

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