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Properties

Label 2-18e2-4.3-c2-0-20
Degree 22
Conductor 324324
Sign −0.0581−0.998i-0.0581 - 0.998i
Analytic cond. 8.828368.82836
Root an. cond. 2.971252.97125
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.45 + 1.37i)2-s + (0.232 + 3.99i)4-s + 4.37·5-s − 2.13i·7-s + (−5.14 + 6.12i)8-s + (6.36 + 6.00i)10-s + 12.6i·11-s + 9.26·13-s + (2.93 − 3.10i)14-s + (−15.8 + 1.85i)16-s + 14.6·17-s + 34.5i·19-s + (1.01 + 17.4i)20-s + (−17.3 + 18.3i)22-s − 40.7i·23-s + ⋯
L(s)  = 1  + (0.727 + 0.686i)2-s + (0.0581 + 0.998i)4-s + 0.874·5-s − 0.305i·7-s + (−0.642 + 0.766i)8-s + (0.636 + 0.600i)10-s + 1.14i·11-s + 0.712·13-s + (0.209 − 0.221i)14-s + (−0.993 + 0.116i)16-s + 0.861·17-s + 1.81i·19-s + (0.0508 + 0.873i)20-s + (−0.787 + 0.834i)22-s − 1.77i·23-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=((−0.0581−0.998i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+1) L(s)=((−0.0581−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −0.0581−0.998i-0.0581 - 0.998i
Analytic conductor: 8.828368.82836
Root analytic conductor: 2.971252.97125
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ324(163,⋅)\chi_{324} (163, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 324, ( :1), −0.0581−0.998i)(2,\ 324,\ (\ :1),\ -0.0581 - 0.998i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.88824+2.00145i1.88824 + 2.00145i
L(12)L(\frac12) ≈\approx 1.88824+2.00145i1.88824 + 2.00145i
L(2)L(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+(−1.45−1.37i)T 1 + (-1.45 - 1.37i)T
3 1 1
good5 1−4.37T+25T2 1 - 4.37T + 25T^{2}
7 1+2.13iT−49T2 1 + 2.13iT - 49T^{2}
11 1−12.6iT−121T2 1 - 12.6iT - 121T^{2}
13 1−9.26T+169T2 1 - 9.26T + 169T^{2}
17 1−14.6T+289T2 1 - 14.6T + 289T^{2}
19 1−34.5iT−361T2 1 - 34.5iT - 361T^{2}
23 1+40.7iT−529T2 1 + 40.7iT - 529T^{2}
29 1−19.0T+841T2 1 - 19.0T + 841T^{2}
31 1+0.993iT−961T2 1 + 0.993iT - 961T^{2}
37 1+66.4T+1.36e3T2 1 + 66.4T + 1.36e3T^{2}
41 1−25.8T+1.68e3T2 1 - 25.8T + 1.68e3T^{2}
43 1−42.1iT−1.84e3T2 1 - 42.1iT - 1.84e3T^{2}
47 1+34.7iT−2.20e3T2 1 + 34.7iT - 2.20e3T^{2}
53 1+12.2T+2.80e3T2 1 + 12.2T + 2.80e3T^{2}
59 1+56.7iT−3.48e3T2 1 + 56.7iT - 3.48e3T^{2}
61 1−73.7T+3.72e3T2 1 - 73.7T + 3.72e3T^{2}
67 1+95.1iT−4.48e3T2 1 + 95.1iT - 4.48e3T^{2}
71 1+75.5iT−5.04e3T2 1 + 75.5iT - 5.04e3T^{2}
73 1+56.7T+5.32e3T2 1 + 56.7T + 5.32e3T^{2}
79 1+74.5iT−6.24e3T2 1 + 74.5iT - 6.24e3T^{2}
83 1+65.4iT−6.88e3T2 1 + 65.4iT - 6.88e3T^{2}
89 1−150.T+7.92e3T2 1 - 150.T + 7.92e3T^{2}
97 1+112.T+9.40e3T2 1 + 112.T + 9.40e3T^{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

−12.12090024968682081935254245669, −10.57297707158069448459228521834, −9.854785030581807334780179564798, −8.592408174335938519109175426896, −7.68220310011742659946544154724, −6.57271440882886449156759966368, −5.81899948362685481559376816154, −4.70625366319570913444676343566, −3.55468903140619532566820123521, −1.95920033568331579770233306318, 1.15731195499420197718237707953, 2.64612403004092162151166691107, 3.72201226965349808853110949283, 5.34339344972764106516134379000, 5.79907859752791450625678988083, 6.98469966510434410691474120148, 8.673146225890591678153792421864, 9.428956038571983120438217875899, 10.41297887800152199954822932100, 11.25783316485745392430178350290

Graph of the ZZ-function along the critical line