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Properties

Label 2-588-7.2-c5-0-7
Degree 22
Conductor 588588
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 − 7.79i)3-s + (23.0 − 39.9i)5-s + (−40.5 + 70.1i)9-s + (315. + 546. i)11-s + 1.07e3·13-s − 415.·15-s + (80.5 + 139. i)17-s + (588. − 1.01e3i)19-s + (−1.08e3 + 1.87e3i)23-s + (499. + 864. i)25-s + 729·27-s − 4.49e3·29-s + (159. + 275. i)31-s + (2.84e3 − 4.91e3i)33-s + (−7.59e3 + 1.31e4i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.412 − 0.714i)5-s + (−0.166 + 0.288i)9-s + (0.786 + 1.36i)11-s + 1.77·13-s − 0.476·15-s + (0.0676 + 0.117i)17-s + (0.373 − 0.647i)19-s + (−0.426 + 0.738i)23-s + (0.159 + 0.276i)25-s + 0.192·27-s − 0.991·29-s + (0.0297 + 0.0515i)31-s + (0.454 − 0.786i)33-s + (−0.911 + 1.57i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ588(373,⋅)\chi_{588} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), 0.605−0.795i)(2,\ 588,\ (\ :5/2),\ 0.605 - 0.795i)

Particular Values

L(3)L(3) ≈\approx 1.9087174831.908717483
L(12)L(\frac12) ≈\approx 1.9087174831.908717483
L(72)L(\frac{7}{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+(4.5+7.79i)T 1 + (4.5 + 7.79i)T
7 1 1
good5 1+(−23.0+39.9i)T+(−1.56e3−2.70e3i)T2 1 + (-23.0 + 39.9i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−315.−546.i)T+(−8.05e4+1.39e5i)T2 1 + (-315. - 546. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−1.07e3T+3.71e5T2 1 - 1.07e3T + 3.71e5T^{2}
17 1+(−80.5−139.i)T+(−7.09e5+1.22e6i)T2 1 + (-80.5 - 139. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−588.+1.01e3i)T+(−1.23e6−2.14e6i)T2 1 + (-588. + 1.01e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(1.08e3−1.87e3i)T+(−3.21e6−5.57e6i)T2 1 + (1.08e3 - 1.87e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+4.49e3T+2.05e7T2 1 + 4.49e3T + 2.05e7T^{2}
31 1+(−159.−275.i)T+(−1.43e7+2.47e7i)T2 1 + (-159. - 275. i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(7.59e3−1.31e4i)T+(−3.46e7−6.00e7i)T2 1 + (7.59e3 - 1.31e4i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+2.05e4T+1.15e8T2 1 + 2.05e4T + 1.15e8T^{2}
43 1+455.T+1.47e8T2 1 + 455.T + 1.47e8T^{2}
47 1+(1.03e4−1.79e4i)T+(−1.14e8−1.98e8i)T2 1 + (1.03e4 - 1.79e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−9.65e3−1.67e4i)T+(−2.09e8+3.62e8i)T2 1 + (-9.65e3 - 1.67e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−3.18e3−5.51e3i)T+(−3.57e8+6.19e8i)T2 1 + (-3.18e3 - 5.51e3i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(2.45e4−4.25e4i)T+(−4.22e8−7.31e8i)T2 1 + (2.45e4 - 4.25e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(1.70e4+2.94e4i)T+(−6.75e8+1.16e9i)T2 1 + (1.70e4 + 2.94e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1−6.29e4T+1.80e9T2 1 - 6.29e4T + 1.80e9T^{2}
73 1+(4.43e3+7.67e3i)T+(−1.03e9+1.79e9i)T2 1 + (4.43e3 + 7.67e3i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(1.72e4−2.98e4i)T+(−1.53e9−2.66e9i)T2 1 + (1.72e4 - 2.98e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−7.04e3T+3.93e9T2 1 - 7.04e3T + 3.93e9T^{2}
89 1+(1.01e4−1.75e4i)T+(−2.79e9−4.83e9i)T2 1 + (1.01e4 - 1.75e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+5.40e4T+8.58e9T2 1 + 5.40e4T + 8.58e9T^{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

−9.921022852659037381891472500301, −9.139568316011311061523957954339, −8.375508764449549254863142463599, −7.25090734595754415004215710410, −6.45157354545787009794112421254, −5.52006917310691460699254781193, −4.56825482032869761728180816621, −3.40804613057975487160854921369, −1.66493122688237954049324871011, −1.26700399564779224186915229426, 0.44926658143144563230536709623, 1.75796981814511933371357216513, 3.38470857199495160400649404749, 3.78725611736600871477000197084, 5.41196985276165885234242519960, 6.14327102457806039356258360864, 6.77168597906570616130625824040, 8.295583238786855921632599363786, 8.852365015772599403343746282036, 9.933931612422096879362135231267

Graph of the ZZ-function along the critical line