Properties

Label 2-588-7.2-c5-0-16
Degree $2$
Conductor $588$
Sign $0.991 + 0.126i$
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 + (3.19 − 5.53i)5-s + (−40.5 + 70.1i)9-s + (−372. − 645. i)11-s − 879.·13-s + 57.5·15-s + (987. + 1.71e3i)17-s + (−484. + 838. i)19-s + (−903. + 1.56e3i)23-s + (1.54e3 + 2.67e3i)25-s − 729·27-s + 547.·29-s + (−3.16e3 − 5.48e3i)31-s + (3.35e3 − 5.81e3i)33-s + (7.31e3 − 1.26e4i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.0571 − 0.0990i)5-s + (−0.166 + 0.288i)9-s + (−0.928 − 1.60i)11-s − 1.44·13-s + 0.0660·15-s + (0.828 + 1.43i)17-s + (−0.307 + 0.533i)19-s + (−0.356 + 0.616i)23-s + (0.493 + 0.854i)25-s − 0.192·27-s + 0.120·29-s + (−0.591 − 1.02i)31-s + (0.536 − 0.928i)33-s + (0.878 − 1.52i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.797869317\)
\(L(\frac12)\) \(\approx\) \(1.797869317\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.19 + 5.53i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (372. + 645. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 879.T + 3.71e5T^{2} \)
17 \( 1 + (-987. - 1.71e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (484. - 838. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (903. - 1.56e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 547.T + 2.05e7T^{2} \)
31 \( 1 + (3.16e3 + 5.48e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-7.31e3 + 1.26e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.49e4T + 1.15e8T^{2} \)
43 \( 1 - 1.66e4T + 1.47e8T^{2} \)
47 \( 1 + (3.82e3 - 6.63e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.19e4 + 2.07e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-7.69e3 - 1.33e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.36e4 + 2.36e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-259. - 449. i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.00e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.28e4 - 5.68e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.96e4 + 3.41e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 3.55e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.43e4 + 9.41e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.41e5T + 8.58e9T^{2} \)
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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

−9.882687258022857403748346771134, −9.097774908365672639911282753337, −8.036835663849095601704901111953, −7.58760164352517093504566315305, −5.90663910026394327586914059808, −5.47945009723920063117436592982, −4.14246115898631583417210820180, −3.22093150029966266512857104489, −2.14402931051756749851783805941, −0.53908508565785287957624845319, 0.71831914637873950106436634432, 2.31679189728408667007625718891, 2.74966455186430124623877266018, 4.55845466882780520401093085428, 5.12345605459749398331479165205, 6.58245213873488405263182907693, 7.39565615086740328900116067892, 7.85037900739144301225080585989, 9.204284015616454977036875286416, 9.866557745232205196604032802681

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