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Properties

Label 2-588-7.2-c5-0-12
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 + (−46.4 + 80.3i)5-s + (−40.5 + 70.1i)9-s + (−70.3 − 121. i)11-s + 1.11e3·13-s + 835.·15-s + (27.4 + 47.5i)17-s + (−855. + 1.48e3i)19-s + (1.64e3 − 2.84e3i)23-s + (−2.74e3 − 4.75e3i)25-s + 729·27-s − 3.79e3·29-s + (−2.42e3 − 4.19e3i)31-s + (−633. + 1.09e3i)33-s + (5.68e3 − 9.84e3i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.830 + 1.43i)5-s + (−0.166 + 0.288i)9-s + (−0.175 − 0.303i)11-s + 1.82·13-s + 0.958·15-s + (0.0230 + 0.0398i)17-s + (−0.543 + 0.942i)19-s + (0.647 − 1.12i)23-s + (−0.878 − 1.52i)25-s + 0.192·27-s − 0.837·29-s + (−0.452 − 0.784i)31-s + (−0.101 + 0.175i)33-s + (0.682 − 1.18i)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 1.4827074661.482707466
L(12)L(\frac12) ≈\approx 1.4827074661.482707466
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+(46.4−80.3i)T+(−1.56e3−2.70e3i)T2 1 + (46.4 - 80.3i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(70.3+121.i)T+(−8.05e4+1.39e5i)T2 1 + (70.3 + 121. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−1.11e3T+3.71e5T2 1 - 1.11e3T + 3.71e5T^{2}
17 1+(−27.4−47.5i)T+(−7.09e5+1.22e6i)T2 1 + (-27.4 - 47.5i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(855.−1.48e3i)T+(−1.23e6−2.14e6i)T2 1 + (855. - 1.48e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−1.64e3+2.84e3i)T+(−3.21e6−5.57e6i)T2 1 + (-1.64e3 + 2.84e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+3.79e3T+2.05e7T2 1 + 3.79e3T + 2.05e7T^{2}
31 1+(2.42e3+4.19e3i)T+(−1.43e7+2.47e7i)T2 1 + (2.42e3 + 4.19e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−5.68e3+9.84e3i)T+(−3.46e7−6.00e7i)T2 1 + (-5.68e3 + 9.84e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−1.03e4T+1.15e8T2 1 - 1.03e4T + 1.15e8T^{2}
43 1−7.13e3T+1.47e8T2 1 - 7.13e3T + 1.47e8T^{2}
47 1+(8.20e3−1.42e4i)T+(−1.14e8−1.98e8i)T2 1 + (8.20e3 - 1.42e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−1.04e4−1.81e4i)T+(−2.09e8+3.62e8i)T2 1 + (-1.04e4 - 1.81e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.81e4+3.13e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.81e4 + 3.13e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−2.47e3+4.28e3i)T+(−4.22e8−7.31e8i)T2 1 + (-2.47e3 + 4.28e3i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−1.14e4−1.98e4i)T+(−6.75e8+1.16e9i)T2 1 + (-1.14e4 - 1.98e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+2.63e4T+1.80e9T2 1 + 2.63e4T + 1.80e9T^{2}
73 1+(−2.76e4−4.79e4i)T+(−1.03e9+1.79e9i)T2 1 + (-2.76e4 - 4.79e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(−2.49e4+4.32e4i)T+(−1.53e9−2.66e9i)T2 1 + (-2.49e4 + 4.32e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+4.48e4T+3.93e9T2 1 + 4.48e4T + 3.93e9T^{2}
89 1+(−6.39e4+1.10e5i)T+(−2.79e9−4.83e9i)T2 1 + (-6.39e4 + 1.10e5i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−6.56e4T+8.58e9T2 1 - 6.56e4T + 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

−10.47860170879590065496709682270, −9.023465834159755314315616098733, −8.025785941214688908660850533677, −7.42377008553166407080261920481, −6.33472450832149523977959188111, −5.93558755208266795060211851269, −4.14937366059269489125663190653, −3.39228616223431254588499237447, −2.25282808466384982199113150622, −0.76584839327575808548131041259, 0.51533397506853286279368896396, 1.46324361554790387047470707127, 3.38358124510480296654621372773, 4.22867755304245865214447485899, 5.03060154365125062957758293725, 5.91949850669519986425083679232, 7.19210613730961670651184978345, 8.294490047036297988266906198400, 8.854283785954001424562194205455, 9.579949037548859817820316451292

Graph of the ZZ-function along the critical line