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Properties

Label 2-588-7.4-c5-0-15
Degree $2$
Conductor $588$
Sign $0.605 + 0.795i$
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 + (−15.9 − 27.6i)5-s + (−40.5 − 70.1i)9-s + (−130. + 225. i)11-s − 769.·13-s + 287.·15-s + (−776. + 1.34e3i)17-s + (375. + 649. i)19-s + (−377. − 653. i)23-s + (1.05e3 − 1.82e3i)25-s + 729·27-s + 6.00e3·29-s + (−3.21e3 + 5.55e3i)31-s + (−1.17e3 − 2.03e3i)33-s + (2.38e3 + 4.13e3i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.285 − 0.495i)5-s + (−0.166 − 0.288i)9-s + (−0.325 + 0.562i)11-s − 1.26·13-s + 0.330·15-s + (−0.651 + 1.12i)17-s + (0.238 + 0.412i)19-s + (−0.148 − 0.257i)23-s + (0.336 − 0.582i)25-s + 0.192·27-s + 1.32·29-s + (−0.599 + 1.03i)31-s + (−0.187 − 0.325i)33-s + (0.286 + 0.496i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 588,\ (\ :5/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8138080183\)
\(L(\frac12)\) \(\approx\) \(0.8138080183\)
\(L(\frac{7}{2})\) 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 \( 1 + (15.9 + 27.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (130. - 225. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 769.T + 3.71e5T^{2} \)
17 \( 1 + (776. - 1.34e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-375. - 649. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (377. + 653. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 6.00e3T + 2.05e7T^{2} \)
31 \( 1 + (3.21e3 - 5.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-2.38e3 - 4.13e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 5.42e3T + 1.15e8T^{2} \)
43 \( 1 + 1.18e4T + 1.47e8T^{2} \)
47 \( 1 + (8.71e3 + 1.50e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.88e4 - 3.26e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.10e4 + 1.91e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.08e3 + 7.07e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (6.50e3 - 1.12e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.23e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.18e4 + 3.77e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.83e4 + 6.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 2.18e4T + 3.93e9T^{2} \)
89 \( 1 + (6.84e4 + 1.18e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 9.30e4T + 8.58e9T^{2} \)
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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

−10.01830358340764411590627262832, −8.884296405336001080684895419057, −8.149229951522144959199136771036, −7.09226250658720736496133429294, −6.09798772785188995060767031164, −4.86497174274568538372822572289, −4.46232482733617553076625153097, −3.09511539833759666465238796610, −1.78406034225996911245922485760, −0.26767343996333052369294636123, 0.70568049398851539419256735121, 2.30236535280910748664620517995, 3.11286854645197789354337787934, 4.60423421140048572668946027780, 5.45420599643476119715017675012, 6.64372872747762649061109860804, 7.28567970380799604998258816587, 8.072530210998777706494762404070, 9.231003961595413095069555566307, 10.04863039081138182279377917370

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