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Properties

Label 2-588-7.4-c3-0-3
Degree 22
Conductor 588588
Sign −0.266−0.963i-0.266 - 0.963i
Analytic cond. 34.693134.6931
Root an. cond. 5.890085.89008
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(588s/2 ΓC(s) L(s)=((−0.266−0.963i) Λ‾(4−s)\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}
Λ(s)=(588s/2 ΓC(s+3/2) L(s)=((−0.266−0.963i) Λ‾(1−s)\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: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: −0.266−0.963i-0.266 - 0.963i
Analytic conductor: 34.693134.6931
Root analytic conductor: 5.890085.89008
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ588(361,⋅)\chi_{588} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :3/2), −0.266−0.963i)(2,\ 588,\ (\ :3/2),\ -0.266 - 0.963i)

Particular Values

L(2)L(2) ≈\approx 1.5978445871.597844587
L(12)L(\frac12) ≈\approx 1.5978445871.597844587
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(−1.5+2.59i)T 1 + (-1.5 + 2.59i)T
7 1 1
good5 1+(−7−12.1i)T+(−62.5+108.i)T2 1 + (-7 - 12.1i)T + (-62.5 + 108. i)T^{2}
11 1+(2−3.46i)T+(−665.5−1.15e3i)T2 1 + (2 - 3.46i)T + (-665.5 - 1.15e3i)T^{2}
13 1+54T+2.19e3T2 1 + 54T + 2.19e3T^{2}
17 1+(7−12.1i)T+(−2.45e3−4.25e3i)T2 1 + (7 - 12.1i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(−46−79.6i)T+(−3.42e3+5.94e3i)T2 1 + (-46 - 79.6i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−76−131.i)T+(−6.08e3+1.05e4i)T2 1 + (-76 - 131. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+106T+2.43e4T2 1 + 106T + 2.43e4T^{2}
31 1+(72−124.i)T+(−1.48e4−2.57e4i)T2 1 + (72 - 124. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(79+136.i)T+(−2.53e4+4.38e4i)T2 1 + (79 + 136. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−390T+6.89e4T2 1 - 390T + 6.89e4T^{2}
43 1+508T+7.95e4T2 1 + 508T + 7.95e4T^{2}
47 1+(264+457.i)T+(−5.19e4+8.99e4i)T2 1 + (264 + 457. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(303−524.i)T+(−7.44e4−1.28e5i)T2 1 + (303 - 524. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(182−315.i)T+(−1.02e5−1.77e5i)T2 1 + (182 - 315. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−339−587.i)T+(−1.13e5+1.96e5i)T2 1 + (-339 - 587. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(422−730.i)T+(−1.50e5−2.60e5i)T2 1 + (422 - 730. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+8T+3.57e5T2 1 + 8T + 3.57e5T^{2}
73 1+(211−365.i)T+(−1.94e5−3.36e5i)T2 1 + (211 - 365. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(192+332.i)T+(−2.46e5+4.26e5i)T2 1 + (192 + 332. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−548T+5.71e5T2 1 - 548T + 5.71e5T^{2}
89 1+(−597−1.03e3i)T+(−3.52e5+6.10e5i)T2 1 + (-597 - 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−1.50e3T+9.12e5T2 1 - 1.50e3T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line