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Properties

Label 2-588-21.17-c3-0-36
Degree $2$
Conductor $588$
Sign $-0.944 + 0.327i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.19 + 0.110i)3-s + (−4.87 − 8.44i)5-s + (26.9 + 1.14i)9-s + (−42.9 − 24.8i)11-s − 2.47i·13-s + (−24.3 − 44.3i)15-s + (24.7 − 42.8i)17-s + (−96.0 + 55.4i)19-s + (−149. + 86.4i)23-s + (15.0 − 25.9i)25-s + (140. + 8.92i)27-s − 134. i·29-s + (−2.18 − 1.26i)31-s + (−220. − 133. i)33-s + (58.5 + 101. i)37-s + ⋯
L(s)  = 1  + (0.999 + 0.0212i)3-s + (−0.435 − 0.754i)5-s + (0.999 + 0.0424i)9-s + (−1.17 − 0.680i)11-s − 0.0528i·13-s + (−0.419 − 0.763i)15-s + (0.353 − 0.611i)17-s + (−1.15 + 0.669i)19-s + (−1.35 + 0.783i)23-s + (0.120 − 0.207i)25-s + (0.997 + 0.0636i)27-s − 0.861i·29-s + (−0.0126 − 0.00730i)31-s + (−1.16 − 0.705i)33-s + (0.260 + 0.450i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.944 + 0.327i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.944 + 0.327i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9678350930\)
\(L(\frac12)\) \(\approx\) \(0.9678350930\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.19 - 0.110i)T \)
7 \( 1 \)
good5 \( 1 + (4.87 + 8.44i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (42.9 + 24.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 2.47iT - 2.19e3T^{2} \)
17 \( 1 + (-24.7 + 42.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (96.0 - 55.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (149. - 86.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 + (2.18 + 1.26i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-58.5 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 160.T + 6.89e4T^{2} \)
43 \( 1 + 442.T + 7.95e4T^{2} \)
47 \( 1 + (155. + 269. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-248. - 143. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (276. - 478. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (504. - 291. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-450. + 780. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 984. iT - 3.57e5T^{2} \)
73 \( 1 + (178. + 102. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (321. + 557. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 351.T + 5.71e5T^{2} \)
89 \( 1 + (-544. - 942. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.36e3iT - 9.12e5T^{2} \)
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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

−9.895563045762564030771420749766, −8.803492936794224344386609207253, −8.117157844115322478262437396583, −7.68676208848077545886925089083, −6.25839114672153235553387059525, −5.07023375258568850913958454375, −4.09408284680883835439194347078, −3.06455762100139569174052060265, −1.83392847608895794550316995923, −0.22690922413670851089975462170, 1.93051914776315126538401725214, 2.85585566713826213778644249335, 3.89349413084463832568968223937, 4.91457068781209871143727535861, 6.44256316892910932143720800110, 7.26039435926489106828267885430, 8.058697925908356013007293744447, 8.723599888409415345695948023933, 10.02674042507941902685225422201, 10.39869661475734357904312523025

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