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Properties

Label 2-270-135.113-c1-0-14
Degree 22
Conductor 270270
Sign −0.505+0.863i-0.505 + 0.863i
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  + (−0.996 − 0.0871i)2-s + (1.12 − 1.31i)3-s + (0.984 + 0.173i)4-s + (−1.30 + 1.81i)5-s + (−1.23 + 1.21i)6-s + (−3.63 − 2.54i)7-s + (−0.965 − 0.258i)8-s + (−0.483 − 2.96i)9-s + (1.46 − 1.69i)10-s + (1.06 − 2.91i)11-s + (1.33 − 1.10i)12-s + (−0.443 − 5.06i)13-s + (3.39 + 2.85i)14-s + (0.922 + 3.76i)15-s + (0.939 + 0.342i)16-s + (−2.90 + 0.778i)17-s + ⋯
L(s)  = 1  + (−0.704 − 0.0616i)2-s + (0.647 − 0.761i)3-s + (0.492 + 0.0868i)4-s + (−0.585 + 0.810i)5-s + (−0.503 + 0.496i)6-s + (−1.37 − 0.961i)7-s + (−0.341 − 0.0915i)8-s + (−0.161 − 0.986i)9-s + (0.462 − 0.534i)10-s + (0.319 − 0.879i)11-s + (0.385 − 0.318i)12-s + (−0.122 − 1.40i)13-s + (0.908 + 0.762i)14-s + (0.238 + 0.971i)15-s + (0.234 + 0.0855i)16-s + (−0.704 + 0.188i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.505+0.863i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((−0.505+0.863i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.505+0.863i-0.505 + 0.863i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(113,⋅)\chi_{270} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), −0.505+0.863i)(2,\ 270,\ (\ :1/2),\ -0.505 + 0.863i)

Particular Values

L(1)L(1) ≈\approx 0.366714−0.639443i0.366714 - 0.639443i
L(12)L(\frac12) ≈\approx 0.366714−0.639443i0.366714 - 0.639443i
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+(0.996+0.0871i)T 1 + (0.996 + 0.0871i)T
3 1+(−1.12+1.31i)T 1 + (-1.12 + 1.31i)T
5 1+(1.30−1.81i)T 1 + (1.30 - 1.81i)T
good7 1+(3.63+2.54i)T+(2.39+6.57i)T2 1 + (3.63 + 2.54i)T + (2.39 + 6.57i)T^{2}
11 1+(−1.06+2.91i)T+(−8.42−7.07i)T2 1 + (-1.06 + 2.91i)T + (-8.42 - 7.07i)T^{2}
13 1+(0.443+5.06i)T+(−12.8+2.25i)T2 1 + (0.443 + 5.06i)T + (-12.8 + 2.25i)T^{2}
17 1+(2.90−0.778i)T+(14.7−8.5i)T2 1 + (2.90 - 0.778i)T + (14.7 - 8.5i)T^{2}
19 1+(−4.83+2.78i)T+(9.5−16.4i)T2 1 + (-4.83 + 2.78i)T + (9.5 - 16.4i)T^{2}
23 1+(−3.26−4.66i)T+(−7.86+21.6i)T2 1 + (-3.26 - 4.66i)T + (-7.86 + 21.6i)T^{2}
29 1+(0.945−0.793i)T+(5.03−28.5i)T2 1 + (0.945 - 0.793i)T + (5.03 - 28.5i)T^{2}
31 1+(0.863−4.89i)T+(−29.1−10.6i)T2 1 + (0.863 - 4.89i)T + (-29.1 - 10.6i)T^{2}
37 1+(0.154+0.578i)T+(−32.0+18.5i)T2 1 + (0.154 + 0.578i)T + (-32.0 + 18.5i)T^{2}
41 1+(−0.230+0.275i)T+(−7.11−40.3i)T2 1 + (-0.230 + 0.275i)T + (-7.11 - 40.3i)T^{2}
43 1+(−0.843−1.80i)T+(−27.6+32.9i)T2 1 + (-0.843 - 1.80i)T + (-27.6 + 32.9i)T^{2}
47 1+(−4.62+6.60i)T+(−16.0−44.1i)T2 1 + (-4.62 + 6.60i)T + (-16.0 - 44.1i)T^{2}
53 1+(5.61−5.61i)T−53iT2 1 + (5.61 - 5.61i)T - 53iT^{2}
59 1+(4.22−1.53i)T+(45.1−37.9i)T2 1 + (4.22 - 1.53i)T + (45.1 - 37.9i)T^{2}
61 1+(−0.293−1.66i)T+(−57.3+20.8i)T2 1 + (-0.293 - 1.66i)T + (-57.3 + 20.8i)T^{2}
67 1+(−8.65+0.757i)T+(65.9−11.6i)T2 1 + (-8.65 + 0.757i)T + (65.9 - 11.6i)T^{2}
71 1+(−4.54−2.62i)T+(35.5+61.4i)T2 1 + (-4.54 - 2.62i)T + (35.5 + 61.4i)T^{2}
73 1+(−0.146+0.545i)T+(−63.2−36.5i)T2 1 + (-0.146 + 0.545i)T + (-63.2 - 36.5i)T^{2}
79 1+(−4.05−4.82i)T+(−13.7+77.7i)T2 1 + (-4.05 - 4.82i)T + (-13.7 + 77.7i)T^{2}
83 1+(−1.20+13.7i)T+(−81.7−14.4i)T2 1 + (-1.20 + 13.7i)T + (-81.7 - 14.4i)T^{2}
89 1+(6.98+12.0i)T+(−44.5+77.0i)T2 1 + (6.98 + 12.0i)T + (-44.5 + 77.0i)T^{2}
97 1+(−13.7+6.41i)T+(62.3−74.3i)T2 1 + (-13.7 + 6.41i)T + (62.3 - 74.3i)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

−11.44253468434214415254077425057, −10.60289693009784208465753270884, −9.654146502114500644527843332868, −8.672441718431812612391816229037, −7.49513533231460737380069443165, −7.07035958879863498947971698097, −6.06385561115320096041875075984, −3.44669836210506434875014242953, −3.05135428409544367883328721200, −0.64716839361139304019030809017, 2.29481323420064726230465107854, 3.75012004561011786154855289481, 4.94305308826949734031113316698, 6.44241991544440203930145196564, 7.59086865030396193554221553271, 8.816699812213119525976020385312, 9.322260505723927557496466631257, 9.798644524393233141831712196004, 11.27339226486821979853534277622, 12.15457775455885021954783689729

Graph of the ZZ-function along the critical line