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Properties

Label 2-270-27.25-c1-0-5
Degree 22
Conductor 270270
Sign 0.139+0.990i0.139 + 0.990i
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 + (−1.54 + 0.789i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (0.509 + 1.65i)6-s + (1.50 − 0.547i)7-s + (−0.5 + 0.866i)8-s + (1.75 − 2.43i)9-s + (−0.5 − 0.866i)10-s + (1.61 + 1.35i)11-s + (1.71 − 0.214i)12-s + (−1.06 − 6.02i)13-s + (−0.278 − 1.57i)14-s + (−0.673 + 1.59i)15-s + (0.766 + 0.642i)16-s + (−1.69 − 2.94i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.890 + 0.455i)3-s + (−0.469 − 0.171i)4-s + (0.342 − 0.287i)5-s + (0.208 + 0.675i)6-s + (0.568 − 0.206i)7-s + (−0.176 + 0.306i)8-s + (0.584 − 0.811i)9-s + (−0.158 − 0.273i)10-s + (0.486 + 0.408i)11-s + (0.496 − 0.0619i)12-s + (−0.294 − 1.67i)13-s + (−0.0743 − 0.421i)14-s + (−0.173 + 0.412i)15-s + (0.191 + 0.160i)16-s + (−0.411 − 0.713i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.139+0.990i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((0.139+0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.139+0.990i0.139 + 0.990i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(241,⋅)\chi_{270} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), 0.139+0.990i)(2,\ 270,\ (\ :1/2),\ 0.139 + 0.990i)

Particular Values

L(1)L(1) ≈\approx 0.804182−0.699032i0.804182 - 0.699032i
L(12)L(\frac12) ≈\approx 0.804182−0.699032i0.804182 - 0.699032i
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+(1.54−0.789i)T 1 + (1.54 - 0.789i)T
5 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
good7 1+(−1.50+0.547i)T+(5.36−4.49i)T2 1 + (-1.50 + 0.547i)T + (5.36 - 4.49i)T^{2}
11 1+(−1.61−1.35i)T+(1.91+10.8i)T2 1 + (-1.61 - 1.35i)T + (1.91 + 10.8i)T^{2}
13 1+(1.06+6.02i)T+(−12.2+4.44i)T2 1 + (1.06 + 6.02i)T + (-12.2 + 4.44i)T^{2}
17 1+(1.69+2.94i)T+(−8.5+14.7i)T2 1 + (1.69 + 2.94i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.99+5.19i)T+(−9.5−16.4i)T2 1 + (-2.99 + 5.19i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.67−0.973i)T+(17.6+14.7i)T2 1 + (-2.67 - 0.973i)T + (17.6 + 14.7i)T^{2}
29 1+(0.145−0.826i)T+(−27.2−9.91i)T2 1 + (0.145 - 0.826i)T + (-27.2 - 9.91i)T^{2}
31 1+(−3.99−1.45i)T+(23.7+19.9i)T2 1 + (-3.99 - 1.45i)T + (23.7 + 19.9i)T^{2}
37 1+(0.457+0.792i)T+(−18.5+32.0i)T2 1 + (0.457 + 0.792i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.88−10.6i)T+(−38.5+14.0i)T2 1 + (-1.88 - 10.6i)T + (-38.5 + 14.0i)T^{2}
43 1+(−3.42−2.87i)T+(7.46+42.3i)T2 1 + (-3.42 - 2.87i)T + (7.46 + 42.3i)T^{2}
47 1+(9.02−3.28i)T+(36.0−30.2i)T2 1 + (9.02 - 3.28i)T + (36.0 - 30.2i)T^{2}
53 1+10.1T+53T2 1 + 10.1T + 53T^{2}
59 1+(−1.98+1.66i)T+(10.2−58.1i)T2 1 + (-1.98 + 1.66i)T + (10.2 - 58.1i)T^{2}
61 1+(−7.40+2.69i)T+(46.7−39.2i)T2 1 + (-7.40 + 2.69i)T + (46.7 - 39.2i)T^{2}
67 1+(1.35+7.65i)T+(−62.9+22.9i)T2 1 + (1.35 + 7.65i)T + (-62.9 + 22.9i)T^{2}
71 1+(−2.95−5.12i)T+(−35.5+61.4i)T2 1 + (-2.95 - 5.12i)T + (-35.5 + 61.4i)T^{2}
73 1+(−5.58+9.67i)T+(−36.5−63.2i)T2 1 + (-5.58 + 9.67i)T + (-36.5 - 63.2i)T^{2}
79 1+(2.55−14.5i)T+(−74.2−27.0i)T2 1 + (2.55 - 14.5i)T + (-74.2 - 27.0i)T^{2}
83 1+(1.27−7.23i)T+(−77.9−28.3i)T2 1 + (1.27 - 7.23i)T + (-77.9 - 28.3i)T^{2}
89 1+(−6.60+11.4i)T+(−44.5−77.0i)T2 1 + (-6.60 + 11.4i)T + (-44.5 - 77.0i)T^{2}
97 1+(−8.13−6.82i)T+(16.8+95.5i)T2 1 + (-8.13 - 6.82i)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

−11.47214093733221175657055707764, −10.98895511370740700754709223714, −9.889785465682975688064960786355, −9.309470678077910702248416516775, −7.86397946064906322353791653833, −6.52438795644776292444955963349, −5.14327868669293424975586907663, −4.70961125774817408239616530748, −3.04564909267897238727864894653, −0.997088449139598775816049075527, 1.77552548845791663131780511744, 4.09539170206211678341421287456, 5.25342565680344954880152961098, 6.25964328373057205479951166033, 6.93661269834803475703183855180, 8.062079373389212385244354046133, 9.166242459017686393668138507052, 10.30664620891472841413794818479, 11.44011836942759950746852108138, 12.00695785301642787122047407511

Graph of the ZZ-function along the critical line