↑

Properties

Label 2-588-21.5-c3-0-22
Degree $2$
Conductor $588$
Sign $0.997 - 0.0633i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 2.59i)3-s + (13.5 − 23.3i)9-s + 91.7i·13-s + (109.5 + 63.2i)19-s + (62.5 + 108. i)25-s − 140. i·27-s + (163.5 − 94.3i)31-s + (161.5 − 279. i)37-s + (238.5 + 413. i)39-s − 71·43-s + 657·57-s + (810 + 467. i)61-s + (63.5 + 109. i)67-s + (−1.05e3 + 608. i)73-s + (562.5 + 324. i)75-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.5 − 0.866i)9-s + 1.95i·13-s + (1.32 + 0.763i)19-s + (0.5 + 0.866i)25-s − 1.00i·27-s + (0.947 − 0.546i)31-s + (0.717 − 1.24i)37-s + (0.979 + 1.69i)39-s − 0.251·43-s + 1.52·57-s + (1.70 + 0.981i)61-s + (0.115 + 0.200i)67-s + (−1.69 + 0.976i)73-s + (0.866 + 0.499i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.881265942\)
\(L(\frac12)\) \(\approx\) \(2.881265942\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.5 + 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 91.7iT - 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-109.5 - 63.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-163.5 + 94.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-161.5 + 279. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 71T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-810 - 467. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-63.5 - 109. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (1.05e3 - 608. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-693.5 + 1.20e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.37e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.998561139831442594067362524674, −9.326174496023494079955359574699, −8.626722009744932672357138763792, −7.54282890221922682289234963078, −6.94152364794610827115624567011, −5.88888591498044158134754085528, −4.45526585299765705162072649143, −3.51144314592946676026291983126, −2.26440688549531601176972248585, −1.19175388556090003212858804359, 0.908492470768639451528875965591, 2.68574186266894372197806186372, 3.30159839267311491291017389863, 4.66991210030904743473878505478, 5.45582441562956184295837161965, 6.83205218331298574174075091957, 7.932290843983537361536899296417, 8.370687232438488450276617280434, 9.520159408655716459250819644105, 10.15189739663851617241347624453

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