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Properties

Label 2-588-7.4-c3-0-3
Degree $2$
Conductor $588$
Sign $-0.266 - 0.963i$
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  + (1.5 − 2.59i)3-s + (7 + 12.1i)5-s + (−4.5 − 7.79i)9-s + (−2 + 3.46i)11-s − 54·13-s + 42·15-s + (−7 + 12.1i)17-s + (46 + 79.6i)19-s + (76 + 131. i)23-s + (−35.5 + 61.4i)25-s − 27·27-s − 106·29-s + (−72 + 124. i)31-s + (6 + 10.3i)33-s + (−79 − 136. i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.626 + 1.08i)5-s + (−0.166 − 0.288i)9-s + (−0.0548 + 0.0949i)11-s − 1.15·13-s + 0.722·15-s + (−0.0998 + 0.172i)17-s + (0.555 + 0.962i)19-s + (0.689 + 1.19i)23-s + (−0.284 + 0.491i)25-s − 0.192·27-s − 0.678·29-s + (−0.417 + 0.722i)31-s + (0.0316 + 0.0548i)33-s + (−0.351 − 0.607i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.597844587\)
\(L(\frac12)\) \(\approx\) \(1.597844587\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-7 - 12.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 54T + 2.19e3T^{2} \)
17 \( 1 + (7 - 12.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46 - 79.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-76 - 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (72 - 124. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (79 + 136. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 390T + 6.89e4T^{2} \)
43 \( 1 + 508T + 7.95e4T^{2} \)
47 \( 1 + (264 + 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (303 - 524. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (182 - 315. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-339 - 587. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (422 - 730. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 8T + 3.57e5T^{2} \)
73 \( 1 + (211 - 365. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (192 + 332. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 548T + 5.71e5T^{2} \)
89 \( 1 + (-597 - 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.50e3T + 9.12e5T^{2} \)
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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

−10.39544955051264773402122700203, −9.786344349011685064078420030413, −8.867858770056329748471689418240, −7.53940011912673906047771272385, −7.17063279344055630539057821341, −6.09617666685913897919284948215, −5.19963457000013538297070462692, −3.58666956563492461284180143393, −2.63513862147770226206191529248, −1.58933746615254908398879569823, 0.42471349382812744528065549461, 1.97505780060961362998975949531, 3.16370526022427972307693199779, 4.82077834610359064778152599675, 4.93896524381757473834629283242, 6.30271300452228773524402339449, 7.48546556745262627845330633035, 8.465088075967671604162130120000, 9.383253152398531299661583557134, 9.657721566231873130730005243020

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