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Properties

Label 2-270-135.122-c1-0-1
Degree 22
Conductor 270270
Sign 0.935−0.354i0.935 - 0.354i
Analytic cond. 2.155962.15596
Root an. cond. 1.468311.46831
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  + (−0.819 − 0.573i)2-s + (−1.69 − 0.364i)3-s + (0.342 + 0.939i)4-s + (−1.52 + 1.63i)5-s + (1.17 + 1.26i)6-s + (−1.90 − 4.08i)7-s + (0.258 − 0.965i)8-s + (2.73 + 1.23i)9-s + (2.18 − 0.470i)10-s + (2.45 + 2.92i)11-s + (−0.236 − 1.71i)12-s + (2.63 + 3.76i)13-s + (−0.783 + 4.44i)14-s + (3.17 − 2.22i)15-s + (−0.766 + 0.642i)16-s + (0.996 + 3.72i)17-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (−0.977 − 0.210i)3-s + (0.171 + 0.469i)4-s + (−0.680 + 0.733i)5-s + (0.480 + 0.518i)6-s + (−0.720 − 1.54i)7-s + (0.0915 − 0.341i)8-s + (0.911 + 0.411i)9-s + (0.691 − 0.148i)10-s + (0.739 + 0.880i)11-s + (−0.0683 − 0.495i)12-s + (0.730 + 1.04i)13-s + (−0.209 + 1.18i)14-s + (0.819 − 0.573i)15-s + (−0.191 + 0.160i)16-s + (0.241 + 0.902i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.935−0.354i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((0.935−0.354i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.935−0.354i0.935 - 0.354i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(257,⋅)\chi_{270} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), 0.935−0.354i)(2,\ 270,\ (\ :1/2),\ 0.935 - 0.354i)

Particular Values

L(1)L(1) ≈\approx 0.564608+0.103476i0.564608 + 0.103476i
L(12)L(\frac12) ≈\approx 0.564608+0.103476i0.564608 + 0.103476i
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+(0.819+0.573i)T 1 + (0.819 + 0.573i)T
3 1+(1.69+0.364i)T 1 + (1.69 + 0.364i)T
5 1+(1.52−1.63i)T 1 + (1.52 - 1.63i)T
good7 1+(1.90+4.08i)T+(−4.49+5.36i)T2 1 + (1.90 + 4.08i)T + (-4.49 + 5.36i)T^{2}
11 1+(−2.45−2.92i)T+(−1.91+10.8i)T2 1 + (-2.45 - 2.92i)T + (-1.91 + 10.8i)T^{2}
13 1+(−2.63−3.76i)T+(−4.44+12.2i)T2 1 + (-2.63 - 3.76i)T + (-4.44 + 12.2i)T^{2}
17 1+(−0.996−3.72i)T+(−14.7+8.5i)T2 1 + (-0.996 - 3.72i)T + (-14.7 + 8.5i)T^{2}
19 1+(−1.79+1.03i)T+(9.5−16.4i)T2 1 + (-1.79 + 1.03i)T + (9.5 - 16.4i)T^{2}
23 1+(−5.25−2.45i)T+(14.7+17.6i)T2 1 + (-5.25 - 2.45i)T + (14.7 + 17.6i)T^{2}
29 1+(−0.501−2.84i)T+(−27.2+9.91i)T2 1 + (-0.501 - 2.84i)T + (-27.2 + 9.91i)T^{2}
31 1+(−2.26+0.825i)T+(23.7−19.9i)T2 1 + (-2.26 + 0.825i)T + (23.7 - 19.9i)T^{2}
37 1+(−5.03+1.34i)T+(32.0−18.5i)T2 1 + (-5.03 + 1.34i)T + (32.0 - 18.5i)T^{2}
41 1+(7.59+1.34i)T+(38.5+14.0i)T2 1 + (7.59 + 1.34i)T + (38.5 + 14.0i)T^{2}
43 1+(0.879−10.0i)T+(−42.3−7.46i)T2 1 + (0.879 - 10.0i)T + (-42.3 - 7.46i)T^{2}
47 1+(2.65−1.23i)T+(30.2−36.0i)T2 1 + (2.65 - 1.23i)T + (30.2 - 36.0i)T^{2}
53 1+(0.405+0.405i)T+53iT2 1 + (0.405 + 0.405i)T + 53iT^{2}
59 1+(6.05+5.07i)T+(10.2+58.1i)T2 1 + (6.05 + 5.07i)T + (10.2 + 58.1i)T^{2}
61 1+(−12.9−4.72i)T+(46.7+39.2i)T2 1 + (-12.9 - 4.72i)T + (46.7 + 39.2i)T^{2}
67 1+(3.68−2.58i)T+(22.9−62.9i)T2 1 + (3.68 - 2.58i)T + (22.9 - 62.9i)T^{2}
71 1+(−9.41−5.43i)T+(35.5+61.4i)T2 1 + (-9.41 - 5.43i)T + (35.5 + 61.4i)T^{2}
73 1+(0.262+0.0703i)T+(63.2+36.5i)T2 1 + (0.262 + 0.0703i)T + (63.2 + 36.5i)T^{2}
79 1+(−10.2+1.80i)T+(74.2−27.0i)T2 1 + (-10.2 + 1.80i)T + (74.2 - 27.0i)T^{2}
83 1+(−0.680+0.972i)T+(−28.3−77.9i)T2 1 + (-0.680 + 0.972i)T + (-28.3 - 77.9i)T^{2}
89 1+(2.08+3.60i)T+(−44.5+77.0i)T2 1 + (2.08 + 3.60i)T + (-44.5 + 77.0i)T^{2}
97 1+(7.69+0.673i)T+(95.5+16.8i)T2 1 + (7.69 + 0.673i)T + (95.5 + 16.8i)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

−11.62981663482025826062867022221, −11.07318790159762480248621333784, −10.27131468055521950175903819977, −9.484003403848063450693345661662, −7.84166198832374700995297623195, −6.85828450558643760160806291748, −6.60280669027369632658134255843, −4.40001295759807702210539383040, −3.58504915641138565615946483265, −1.25429788771716571724830519055, 0.73740707082981341139838875428, 3.33859481506123119800096603759, 5.10258763699954669024175503320, 5.80351433875033082320738843711, 6.74563281126892795301774389895, 8.209441982156458413987937125364, 8.968074161693609184329864766094, 9.772838344571540071254130075673, 11.06979328757383880172274657849, 11.81265905939190201359469318370

Graph of the ZZ-function along the critical line