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Properties

Label 2-588-7.4-c3-0-2
Degree $2$
Conductor $588$
Sign $-0.991 + 0.126i$
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 + (9 + 15.5i)5-s + (−4.5 − 7.79i)9-s + (−18 + 31.1i)11-s − 10·13-s − 54·15-s + (−9 + 15.5i)17-s + (50 + 86.6i)19-s + (−36 − 62.3i)23-s + (−99.5 + 172. i)25-s + 27·27-s − 234·29-s + (8 − 13.8i)31-s + (−54 − 93.5i)33-s + (113 + 195. i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.804 + 1.39i)5-s + (−0.166 − 0.288i)9-s + (−0.493 + 0.854i)11-s − 0.213·13-s − 0.929·15-s + (−0.128 + 0.222i)17-s + (0.603 + 1.04i)19-s + (−0.326 − 0.565i)23-s + (−0.796 + 1.37i)25-s + 0.192·27-s − 1.49·29-s + (0.0463 − 0.0802i)31-s + (−0.284 − 0.493i)33-s + (0.502 + 0.869i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(34.6931\)
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: \((2,\ 588,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.210920597\)
\(L(\frac12)\) \(\approx\) \(1.210920597\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9 - 15.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 10T + 2.19e3T^{2} \)
17 \( 1 + (9 - 15.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-50 - 86.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (36 + 62.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 234T + 2.43e4T^{2} \)
31 \( 1 + (-8 + 13.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-113 - 195. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 90T + 6.89e4T^{2} \)
43 \( 1 - 452T + 7.95e4T^{2} \)
47 \( 1 + (216 + 374. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (207 - 358. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-342 + 592. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (211 + 365. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (166 - 287. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 360T + 3.57e5T^{2} \)
73 \( 1 + (13 - 22.5i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (256 + 443. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.18e3T + 5.71e5T^{2} \)
89 \( 1 + (-315 - 545. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−10.59236705914163191101140181612, −9.970807271058929252562197325529, −9.415584497361670185830379805312, −7.933977108448886271995466932124, −7.08031246311657906906331551528, −6.15644539849692277135779681699, −5.39467151298850458275652594533, −4.10356131959568091775513434879, −2.93876774037124632334990802024, −1.90393650712032757005193283203, 0.36033358291368580141742077570, 1.41597341145097776431103789488, 2.68847853078401610731931991498, 4.36217978134752729577214306045, 5.47677257733413799724981435660, 5.80806314355480198686974301067, 7.21039550208104507096014507223, 8.090434469174062365152887847078, 9.097861086764725171420017413128, 9.542727893992194909940415398691

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