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Properties

Label 2-675-27.13-c1-0-23
Degree 22
Conductor 675675
Sign −0.0581−0.998i-0.0581 - 0.998i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (0.233 + 1.32i)2-s + (1.70 + 0.300i)3-s + (0.173 − 0.0632i)4-s + 2.33i·6-s + (0.652 + 0.237i)7-s + (1.47 + 2.54i)8-s + (2.81 + 1.02i)9-s + (−3.52 + 2.95i)11-s + (0.315 − 0.0555i)12-s + (−0.245 + 1.39i)13-s + (−0.162 + 0.921i)14-s + (−2.75 + 2.31i)16-s + (1.93 − 3.35i)17-s + (−0.701 + 3.98i)18-s + (−3.53 − 6.11i)19-s + ⋯
L(s)  = 1  + (0.165 + 0.938i)2-s + (0.984 + 0.173i)3-s + (0.0868 − 0.0316i)4-s + 0.952i·6-s + (0.246 + 0.0897i)7-s + (0.520 + 0.901i)8-s + (0.939 + 0.342i)9-s + (−1.06 + 0.890i)11-s + (0.0909 − 0.0160i)12-s + (−0.0679 + 0.385i)13-s + (−0.0434 + 0.246i)14-s + (−0.688 + 0.577i)16-s + (0.470 − 0.814i)17-s + (−0.165 + 0.938i)18-s + (−0.810 − 1.40i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((−0.0581−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((−0.0581−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −0.0581−0.998i-0.0581 - 0.998i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(526,⋅)\chi_{675} (526, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), −0.0581−0.998i)(2,\ 675,\ (\ :1/2),\ -0.0581 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.76391+1.86964i1.76391 + 1.86964i
L(12)L(\frac12) ≈\approx 1.76391+1.86964i1.76391 + 1.86964i
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)
bad3 1+(−1.70−0.300i)T 1 + (-1.70 - 0.300i)T
5 1 1
good2 1+(−0.233−1.32i)T+(−1.87+0.684i)T2 1 + (-0.233 - 1.32i)T + (-1.87 + 0.684i)T^{2}
7 1+(−0.652−0.237i)T+(5.36+4.49i)T2 1 + (-0.652 - 0.237i)T + (5.36 + 4.49i)T^{2}
11 1+(3.52−2.95i)T+(1.91−10.8i)T2 1 + (3.52 - 2.95i)T + (1.91 - 10.8i)T^{2}
13 1+(0.245−1.39i)T+(−12.2−4.44i)T2 1 + (0.245 - 1.39i)T + (-12.2 - 4.44i)T^{2}
17 1+(−1.93+3.35i)T+(−8.5−14.7i)T2 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.53+6.11i)T+(−9.5+16.4i)T2 1 + (3.53 + 6.11i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.59+1.30i)T+(17.6−14.7i)T2 1 + (-3.59 + 1.30i)T + (17.6 - 14.7i)T^{2}
29 1+(−0.851−4.82i)T+(−27.2+9.91i)T2 1 + (-0.851 - 4.82i)T + (-27.2 + 9.91i)T^{2}
31 1+(−0.786+0.286i)T+(23.7−19.9i)T2 1 + (-0.786 + 0.286i)T + (23.7 - 19.9i)T^{2}
37 1+(−3.99+6.91i)T+(−18.5−32.0i)T2 1 + (-3.99 + 6.91i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.36−7.74i)T+(−38.5−14.0i)T2 1 + (1.36 - 7.74i)T + (-38.5 - 14.0i)T^{2}
43 1+(1.59−1.33i)T+(7.46−42.3i)T2 1 + (1.59 - 1.33i)T + (7.46 - 42.3i)T^{2}
47 1+(6.46+2.35i)T+(36.0+30.2i)T2 1 + (6.46 + 2.35i)T + (36.0 + 30.2i)T^{2}
53 1−3.05T+53T2 1 - 3.05T + 53T^{2}
59 1+(6.82+5.72i)T+(10.2+58.1i)T2 1 + (6.82 + 5.72i)T + (10.2 + 58.1i)T^{2}
61 1+(−8.12−2.95i)T+(46.7+39.2i)T2 1 + (-8.12 - 2.95i)T + (46.7 + 39.2i)T^{2}
67 1+(−1.64+9.30i)T+(−62.9−22.9i)T2 1 + (-1.64 + 9.30i)T + (-62.9 - 22.9i)T^{2}
71 1+(−2.90+5.02i)T+(−35.5−61.4i)T2 1 + (-2.90 + 5.02i)T + (-35.5 - 61.4i)T^{2}
73 1+(2.70+4.68i)T+(−36.5+63.2i)T2 1 + (2.70 + 4.68i)T + (-36.5 + 63.2i)T^{2}
79 1+(2.27+12.9i)T+(−74.2+27.0i)T2 1 + (2.27 + 12.9i)T + (-74.2 + 27.0i)T^{2}
83 1+(0.197+1.11i)T+(−77.9+28.3i)T2 1 + (0.197 + 1.11i)T + (-77.9 + 28.3i)T^{2}
89 1+(0.368+0.637i)T+(−44.5+77.0i)T2 1 + (0.368 + 0.637i)T + (-44.5 + 77.0i)T^{2}
97 1+(6.39−5.36i)T+(16.8−95.5i)T2 1 + (6.39 - 5.36i)T + (16.8 - 95.5i)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

−10.63248009754263053023952847854, −9.678075685951945999896880585349, −8.798111116354010284389326469298, −7.937671794826592956222124819089, −7.25750403905601093116492320312, −6.58843120619302984898023133019, −5.01371616124218810612299370834, −4.70309415865211164949791064397, −2.92924545213226361576612813480, −2.01852628781566488668008560491, 1.34166349132634338594485166629, 2.52839805020763235455903558379, 3.35693026671192425824738746930, 4.24213228907600621135750554879, 5.71694183588434359539889084104, 6.88589316824613836655539426127, 8.098219047709833425723677865879, 8.221675208966312163814709902007, 9.722979154780473527885539840475, 10.33278592553543956286967106697

Graph of the ZZ-function along the critical line