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Properties

Label 2-588-7.4-c3-0-16
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 + (−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

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

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

\(L(2)\) \(\approx\) \(0.2901319911\)
\(L(\frac12)\) \(\approx\) \(0.2901319911\)
\(L(\frac{5}{2})\) 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

−9.747689147172184987445118173993, −8.946876767163782537914800335277, −8.327907883784595903889668106389, −7.24204085428413570417077202273, −6.04700679747593840765199914588, −5.05862910910488125454326118382, −4.28975681911657657082182059729, −3.27192605291385878575052164001, −1.38160132855859268099628637257, −0.093945551802059977620418330843, 1.56156187896232319924575176752, 3.02702136813274256252712982239, 3.89745433927964674163016213857, 5.32396337660189184897287211027, 6.52315648564995398553627961534, 6.87520399321193183965056400294, 8.109095036545003572351853855640, 8.629662634215789331548687531877, 10.22113591129320116489378126048, 10.71007958256415144953456624160

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