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Properties

Label 2-588-21.2-c2-0-14
Degree $2$
Conductor $588$
Sign $0.374 - 0.927i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.178 + 2.99i)3-s + (5.28 + 3.05i)5-s + (−8.93 − 1.07i)9-s + (14.9 − 8.63i)11-s + 19.2·13-s + (−10.0 + 15.2i)15-s + (17.8 − 10.3i)17-s + (−6.09 + 10.5i)19-s + (16.8 + 9.70i)23-s + (6.12 + 10.6i)25-s + (4.80 − 26.5i)27-s − 3.24i·29-s + (4.24 + 7.34i)31-s + (23.1 + 46.3i)33-s + (−33.8 + 58.6i)37-s + ⋯
L(s)  = 1  + (−0.0595 + 0.998i)3-s + (1.05 + 0.610i)5-s + (−0.992 − 0.118i)9-s + (1.35 − 0.784i)11-s + 1.48·13-s + (−0.672 + 1.01i)15-s + (1.04 − 0.605i)17-s + (−0.320 + 0.555i)19-s + (0.730 + 0.421i)23-s + (0.244 + 0.424i)25-s + (0.177 − 0.984i)27-s − 0.111i·29-s + (0.136 + 0.237i)31-s + (0.702 + 1.40i)33-s + (−0.915 + 1.58i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.374 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.413008859\)
\(L(\frac12)\) \(\approx\) \(2.413008859\)
\(L(2)\) 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 \( 1 + (0.178 - 2.99i)T \)
7 \( 1 \)
good5 \( 1 + (-5.28 - 3.05i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-14.9 + 8.63i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 19.2T + 169T^{2} \)
17 \( 1 + (-17.8 + 10.3i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.09 - 10.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.8 - 9.70i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 3.24iT - 841T^{2} \)
31 \( 1 + (-4.24 - 7.34i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (33.8 - 58.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 55.6iT - 1.68e3T^{2} \)
43 \( 1 + 49.7T + 1.84e3T^{2} \)
47 \( 1 + (39.6 + 22.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-28.9 + 16.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (58.8 - 33.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (9.10 - 15.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (33 + 57.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 93.8iT - 5.04e3T^{2} \)
73 \( 1 + (3.88 + 6.73i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (52.4 - 90.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 51.1iT - 6.88e3T^{2} \)
89 \( 1 + (-130. - 75.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 188.T + 9.40e3T^{2} \)
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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

−10.50636567258363532233328134152, −9.885772616863943377879867839033, −9.018453254563797149123528177411, −8.371604711171194908441262862871, −6.70787754743888070824516269045, −6.05198717529133500109030795535, −5.25948882147221010955860688650, −3.77377862804510225798460838837, −3.13706387331415662879899850167, −1.37626141168481101690675811701, 1.17361988574359094968904346663, 1.81748681847187331959357361584, 3.45388473764476948296601665543, 4.89630237131988670904822919337, 6.06265473449711792516252350274, 6.45886916980739187280115312140, 7.61010846140840608582535722797, 8.765551471854305064441125094412, 9.155339866674062573762175499891, 10.30921630747933763977479646428

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