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Properties

Label 2-5292-63.59-c1-0-8
Degree 22
Conductor 52925292
Sign −0.827−0.562i-0.827 - 0.562i
Analytic cond. 42.256842.2568
Root an. cond. 6.500526.50052
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0764·5-s + 5.38i·11-s + (4.60 + 2.65i)13-s + (−1.89 + 3.27i)17-s + (4.33 − 2.50i)19-s + 2.33i·23-s − 4.99·25-s + (−8.84 + 5.10i)29-s + (−4.97 + 2.87i)31-s + (0.354 + 0.613i)37-s + (3.29 − 5.71i)41-s + (0.716 + 1.24i)43-s + (−1.46 + 2.53i)47-s + (−10.4 − 6.05i)53-s + 0.411i·55-s + ⋯
L(s)  = 1  + 0.0341·5-s + 1.62i·11-s + (1.27 + 0.737i)13-s + (−0.458 + 0.794i)17-s + (0.995 − 0.574i)19-s + 0.487i·23-s − 0.998·25-s + (−1.64 + 0.948i)29-s + (−0.893 + 0.516i)31-s + (0.0582 + 0.100i)37-s + (0.515 − 0.892i)41-s + (0.109 + 0.189i)43-s + (−0.213 + 0.369i)47-s + (−1.44 − 0.831i)53-s + 0.0554i·55-s + ⋯

Functional equation

Λ(s)=(5292s/2 ΓC(s) L(s)=((−0.827−0.562i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5292s/2 ΓC(s+1/2) L(s)=((−0.827−0.562i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52925292    =    22⋅33⋅722^{2} \cdot 3^{3} \cdot 7^{2}
Sign: −0.827−0.562i-0.827 - 0.562i
Analytic conductor: 42.256842.2568
Root analytic conductor: 6.500526.50052
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5292(2285,⋅)\chi_{5292} (2285, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5292, ( :1/2), −0.827−0.562i)(2,\ 5292,\ (\ :1/2),\ -0.827 - 0.562i)

Particular Values

L(1)L(1) ≈\approx 1.1750190191.175019019
L(12)L(\frac12) ≈\approx 1.1750190191.175019019
L(32)L(\frac{3}{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
7 1 1
good5 1−0.0764T+5T2 1 - 0.0764T + 5T^{2}
11 1−5.38iT−11T2 1 - 5.38iT - 11T^{2}
13 1+(−4.60−2.65i)T+(6.5+11.2i)T2 1 + (-4.60 - 2.65i)T + (6.5 + 11.2i)T^{2}
17 1+(1.89−3.27i)T+(−8.5−14.7i)T2 1 + (1.89 - 3.27i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.33+2.50i)T+(9.5−16.4i)T2 1 + (-4.33 + 2.50i)T + (9.5 - 16.4i)T^{2}
23 1−2.33iT−23T2 1 - 2.33iT - 23T^{2}
29 1+(8.84−5.10i)T+(14.5−25.1i)T2 1 + (8.84 - 5.10i)T + (14.5 - 25.1i)T^{2}
31 1+(4.97−2.87i)T+(15.5−26.8i)T2 1 + (4.97 - 2.87i)T + (15.5 - 26.8i)T^{2}
37 1+(−0.354−0.613i)T+(−18.5+32.0i)T2 1 + (-0.354 - 0.613i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.29+5.71i)T+(−20.5−35.5i)T2 1 + (-3.29 + 5.71i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.716−1.24i)T+(−21.5+37.2i)T2 1 + (-0.716 - 1.24i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.46−2.53i)T+(−23.5−40.7i)T2 1 + (1.46 - 2.53i)T + (-23.5 - 40.7i)T^{2}
53 1+(10.4+6.05i)T+(26.5+45.8i)T2 1 + (10.4 + 6.05i)T + (26.5 + 45.8i)T^{2}
59 1+(0.289+0.502i)T+(−29.5+51.0i)T2 1 + (0.289 + 0.502i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.40−1.38i)T+(30.5+52.8i)T2 1 + (-2.40 - 1.38i)T + (30.5 + 52.8i)T^{2}
67 1+(2.63+4.56i)T+(−33.5+58.0i)T2 1 + (2.63 + 4.56i)T + (-33.5 + 58.0i)T^{2}
71 1+3.32iT−71T2 1 + 3.32iT - 71T^{2}
73 1+(−6.17−3.56i)T+(36.5+63.2i)T2 1 + (-6.17 - 3.56i)T + (36.5 + 63.2i)T^{2}
79 1+(0.469−0.812i)T+(−39.5−68.4i)T2 1 + (0.469 - 0.812i)T + (-39.5 - 68.4i)T^{2}
83 1+(6.49+11.2i)T+(−41.5+71.8i)T2 1 + (6.49 + 11.2i)T + (-41.5 + 71.8i)T^{2}
89 1+(1.51+2.62i)T+(−44.5+77.0i)T2 1 + (1.51 + 2.62i)T + (-44.5 + 77.0i)T^{2}
97 1+(6.18−3.56i)T+(48.5−84.0i)T2 1 + (6.18 - 3.56i)T + (48.5 - 84.0i)T^{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

−8.555766786378129487542414366684, −7.55790192560232362731514855055, −7.18764757746605185563282928889, −6.36547650252168320509944346501, −5.60199765040787179534138021784, −4.80909969725965630704984498238, −3.97913058829842787392150343228, −3.38959561653394585902775666418, −1.97226771500967756503373466578, −1.55449456162384735209462609721, 0.30711599072691271137574736038, 1.32403285945678071695111194215, 2.57202407377878418198170846704, 3.48233264572825387147595725802, 3.92305852948332194401132331175, 5.17077685077169775796781530140, 5.95864385693060038453443318527, 6.08890424144124219780425070068, 7.36405838061869091860387575451, 7.934760826136128835151267243822

Graph of the ZZ-function along the critical line