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Properties

Label 2-270-27.13-c1-0-5
Degree 22
Conductor 270270
Sign 0.973−0.230i0.973 - 0.230i
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  + (−0.173 − 0.984i)2-s + (1.11 + 1.32i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s + (1.11 − 1.32i)6-s + (0.652 + 0.237i)7-s + (0.5 + 0.866i)8-s + (−0.520 + 2.95i)9-s + (0.5 − 0.866i)10-s + (1.62 − 1.36i)11-s + (−1.49 − 0.866i)12-s + (−0.532 + 3.01i)13-s + (0.120 − 0.684i)14-s + 1.73i·15-s + (0.766 − 0.642i)16-s + (0.826 − 1.43i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.642 + 0.766i)3-s + (−0.469 + 0.171i)4-s + (0.342 + 0.287i)5-s + (0.454 − 0.541i)6-s + (0.246 + 0.0897i)7-s + (0.176 + 0.306i)8-s + (−0.173 + 0.984i)9-s + (0.158 − 0.273i)10-s + (0.489 − 0.410i)11-s + (−0.433 − 0.249i)12-s + (−0.147 + 0.836i)13-s + (0.0322 − 0.182i)14-s + 0.447i·15-s + (0.191 − 0.160i)16-s + (0.200 − 0.347i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.973−0.230i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((0.973−0.230i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.973−0.230i0.973 - 0.230i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(121,⋅)\chi_{270} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), 0.973−0.230i)(2,\ 270,\ (\ :1/2),\ 0.973 - 0.230i)

Particular Values

L(1)L(1) ≈\approx 1.48561+0.173643i1.48561 + 0.173643i
L(12)L(\frac12) ≈\approx 1.48561+0.173643i1.48561 + 0.173643i
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+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
3 1+(−1.11−1.32i)T 1 + (-1.11 - 1.32i)T
5 1+(−0.766−0.642i)T 1 + (-0.766 - 0.642i)T
good7 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+(−1.62+1.36i)T+(1.91−10.8i)T2 1 + (-1.62 + 1.36i)T + (1.91 - 10.8i)T^{2}
13 1+(0.532−3.01i)T+(−12.2−4.44i)T2 1 + (0.532 - 3.01i)T + (-12.2 - 4.44i)T^{2}
17 1+(−0.826+1.43i)T+(−8.5−14.7i)T2 1 + (-0.826 + 1.43i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.29−2.24i)T+(−9.5+16.4i)T2 1 + (-1.29 - 2.24i)T + (-9.5 + 16.4i)T^{2}
23 1+(−4.18+1.52i)T+(17.6−14.7i)T2 1 + (-4.18 + 1.52i)T + (17.6 - 14.7i)T^{2}
29 1+(1.49+8.45i)T+(−27.2+9.91i)T2 1 + (1.49 + 8.45i)T + (-27.2 + 9.91i)T^{2}
31 1+(6.06−2.20i)T+(23.7−19.9i)T2 1 + (6.06 - 2.20i)T + (23.7 - 19.9i)T^{2}
37 1+(−2+3.46i)T+(−18.5−32.0i)T2 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.12+6.36i)T+(−38.5−14.0i)T2 1 + (-1.12 + 6.36i)T + (-38.5 - 14.0i)T^{2}
43 1+(0.450−0.378i)T+(7.46−42.3i)T2 1 + (0.450 - 0.378i)T + (7.46 - 42.3i)T^{2}
47 1+(4.75+1.73i)T+(36.0+30.2i)T2 1 + (4.75 + 1.73i)T + (36.0 + 30.2i)T^{2}
53 1−6.82T+53T2 1 - 6.82T + 53T^{2}
59 1+(3.98+3.34i)T+(10.2+58.1i)T2 1 + (3.98 + 3.34i)T + (10.2 + 58.1i)T^{2}
61 1+(10.8+3.93i)T+(46.7+39.2i)T2 1 + (10.8 + 3.93i)T + (46.7 + 39.2i)T^{2}
67 1+(0.774−4.39i)T+(−62.9−22.9i)T2 1 + (0.774 - 4.39i)T + (-62.9 - 22.9i)T^{2}
71 1+(5.36−9.30i)T+(−35.5−61.4i)T2 1 + (5.36 - 9.30i)T + (-35.5 - 61.4i)T^{2}
73 1+(−3.86−6.69i)T+(−36.5+63.2i)T2 1 + (-3.86 - 6.69i)T + (-36.5 + 63.2i)T^{2}
79 1+(0.0837+0.475i)T+(−74.2+27.0i)T2 1 + (0.0837 + 0.475i)T + (-74.2 + 27.0i)T^{2}
83 1+(−1.45−8.26i)T+(−77.9+28.3i)T2 1 + (-1.45 - 8.26i)T + (-77.9 + 28.3i)T^{2}
89 1+(6.56+11.3i)T+(−44.5+77.0i)T2 1 + (6.56 + 11.3i)T + (-44.5 + 77.0i)T^{2}
97 1+(−12.0+10.1i)T+(16.8−95.5i)T2 1 + (-12.0 + 10.1i)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

−11.64490988789608411117164272631, −11.00970446222121744802497820541, −9.962466408781774087529006805449, −9.287797109024783468792359933405, −8.470322369710060792398018905120, −7.24483173864364016964988217197, −5.64698137664089888496263871125, −4.40200372634762482763585610349, −3.33027061387772200000394541528, −2.02600829523625448412132962464, 1.40202267496782222944332189169, 3.23016217992112470000019674400, 4.86581299157743409332985341204, 6.05576889320444128321332552137, 7.13157446775848738263547794286, 7.86547872363994154166795798962, 8.922183599886993345332464540376, 9.559018872723210795651183712645, 10.88870956749795405660416151538, 12.22281388126373103034615810529

Graph of the ZZ-function along the critical line