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Properties

Label 2-588-7.4-c7-0-18
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 + (−50 − 86.6i)5-s + (−364.5 − 631. i)9-s + (−1.38e3 + 2.40e3i)11-s − 3.29e3·13-s − 2.70e3·15-s + (−2.95e3 + 5.10e3i)17-s + (−3.32e3 − 5.75e3i)19-s + (−991 − 1.71e3i)23-s + (3.40e4 − 5.89e4i)25-s − 1.96e4·27-s − 2.08e5·29-s + (5.88e4 − 1.02e5i)31-s + (3.74e4 + 6.48e4i)33-s + (1.67e5 + 2.90e5i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.178 − 0.309i)5-s + (−0.166 − 0.288i)9-s + (−0.314 + 0.544i)11-s − 0.415·13-s − 0.206·15-s + (−0.145 + 0.252i)17-s + (−0.111 − 0.192i)19-s + (−0.0169 − 0.0294i)23-s + (0.435 − 0.755i)25-s − 0.192·27-s − 1.58·29-s + (0.355 − 0.615i)31-s + (0.181 + 0.314i)33-s + (0.544 + 0.943i)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

\(L(4)\) \(\approx\) \(1.674718660\)
\(L(\frac12)\) \(\approx\) \(1.674718660\)
\(L(\frac{9}{2})\) 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 \( 1 + (50 + 86.6i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (1.38e3 - 2.40e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 3.29e3T + 6.27e7T^{2} \)
17 \( 1 + (2.95e3 - 5.10e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (3.32e3 + 5.75e3i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (991 + 1.71e3i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + 2.08e5T + 1.72e10T^{2} \)
31 \( 1 + (-5.88e4 + 1.02e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-1.67e5 - 2.90e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 2.65e5T + 1.94e11T^{2} \)
43 \( 1 + 9.32e4T + 2.71e11T^{2} \)
47 \( 1 + (-3.28e5 - 5.69e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-3.04e5 + 5.27e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-2.68e5 + 4.64e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-8.98e5 - 1.55e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.06e6 - 1.83e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 1.19e6T + 9.09e12T^{2} \)
73 \( 1 + (5.28e5 - 9.14e5i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (4.99e5 + 8.64e5i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 3.89e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.31e6 - 4.00e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 1.52e7T + 8.07e13T^{2} \)
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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

−9.526310789008828306810841051286, −8.616739772922722157718291234629, −7.81551376058665930169679895286, −7.05102207197775877800884172710, −6.06285218555691671597483061057, −4.95749975630065742813192181952, −4.02602473199963266099005443967, −2.75950054500881910774087486123, −1.85958056382846429834314100631, −0.67074930057982823709831660355, 0.41759672478793209281249335355, 1.92356497833029215926701013168, 3.02111947435136813503268888911, 3.82208079131516615931526356892, 4.96147929170139513697898724779, 5.80497468872726357241131205572, 7.02072238307116102495631711165, 7.79158005181949934800826407394, 8.785263276893203329811772806620, 9.495818987264873884522411299940

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