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Properties

Label 2-588-21.17-c3-0-0
Degree $2$
Conductor $588$
Sign $-0.494 + 0.869i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.30 + 2.90i)3-s + (7.41 + 12.8i)5-s + (10.1 − 25.0i)9-s + (0.589 + 0.340i)11-s − 52.4i·13-s + (−69.2 − 33.7i)15-s + (−51.6 + 89.5i)17-s + (−86.7 + 50.0i)19-s + (−3.63 + 2.10i)23-s + (−47.4 + 82.2i)25-s + (29.2 + 137. i)27-s + 53.5i·29-s + (−235. − 136. i)31-s + (−3.52 + 0.246i)33-s + (25.2 + 43.7i)37-s + ⋯
L(s)  = 1  + (−0.828 + 0.559i)3-s + (0.663 + 1.14i)5-s + (0.374 − 0.927i)9-s + (0.0161 + 0.00932i)11-s − 1.11i·13-s + (−1.19 − 0.581i)15-s + (−0.737 + 1.27i)17-s + (−1.04 + 0.604i)19-s + (−0.0329 + 0.0190i)23-s + (−0.379 + 0.657i)25-s + (0.208 + 0.978i)27-s + 0.342i·29-s + (−1.36 − 0.789i)31-s + (−0.0185 + 0.00130i)33-s + (0.112 + 0.194i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.494 + 0.869i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.494 + 0.869i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.02782328035\)
\(L(\frac12)\) \(\approx\) \(0.02782328035\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.30 - 2.90i)T \)
7 \( 1 \)
good5 \( 1 + (-7.41 - 12.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-0.589 - 0.340i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 52.4iT - 2.19e3T^{2} \)
17 \( 1 + (51.6 - 89.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (86.7 - 50.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (3.63 - 2.10i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 53.5iT - 2.43e4T^{2} \)
31 \( 1 + (235. + 136. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-25.2 - 43.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 318.T + 6.89e4T^{2} \)
43 \( 1 - 168.T + 7.95e4T^{2} \)
47 \( 1 + (313. + 543. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (625. + 361. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (263. - 455. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (205. - 118. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (149. - 258. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 379. iT - 3.57e5T^{2} \)
73 \( 1 + (-559. - 323. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (211. + 366. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 391.T + 5.71e5T^{2} \)
89 \( 1 + (316. + 548. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.43e3iT - 9.12e5T^{2} \)
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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

−10.65414229085901044170556062000, −10.35692981681841412678313865708, −9.405575388223484334484367536141, −8.252059216209292128245221323073, −7.03836083273951757053736862512, −6.14710125974731445838422353288, −5.68386691490705692848163355083, −4.29757527898086076468692559849, −3.28041810946710419180392025230, −1.90123476800795316048489145503, 0.009080615644824276390356472534, 1.31048670312339515996571748062, 2.31975359343653622707411521072, 4.45529414806345712111479576585, 4.97575645461206841702847975112, 6.07807442023222805648126444461, 6.81394813665448587817489850135, 7.84803162813262372029875560793, 9.135697009892796924871805076537, 9.376649361935572158517001059366

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