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Properties

Label 2-18e2-4.3-c2-0-36
Degree 22
Conductor 324324
Sign −0.547+0.836i-0.547 + 0.836i
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  + (−1.75 + 0.951i)2-s + (2.19 − 3.34i)4-s + 1.86·5-s − 11.3i·7-s + (−0.672 + 7.97i)8-s + (−3.27 + 1.77i)10-s − 5.87i·11-s − 13.9·13-s + (10.7 + 19.9i)14-s + (−6.39 − 14.6i)16-s − 11.0·17-s + 9.34i·19-s + (4.08 − 6.23i)20-s + (5.58 + 10.3i)22-s + 30.4i·23-s + ⋯
L(s)  = 1  + (−0.879 + 0.475i)2-s + (0.547 − 0.836i)4-s + 0.372·5-s − 1.61i·7-s + (−0.0840 + 0.996i)8-s + (−0.327 + 0.177i)10-s − 0.533i·11-s − 1.07·13-s + (0.768 + 1.42i)14-s + (−0.399 − 0.916i)16-s − 0.651·17-s + 0.491i·19-s + (0.204 − 0.311i)20-s + (0.253 + 0.469i)22-s + 1.32i·23-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=((−0.547+0.836i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+1) L(s)=((−0.547+0.836i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −0.547+0.836i-0.547 + 0.836i
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), −0.547+0.836i)(2,\ 324,\ (\ :1),\ -0.547 + 0.836i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.266629−0.493259i0.266629 - 0.493259i
L(12)L(\frac12) ≈\approx 0.266629−0.493259i0.266629 - 0.493259i
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+(1.75−0.951i)T 1 + (1.75 - 0.951i)T
3 1 1
good5 1−1.86T+25T2 1 - 1.86T + 25T^{2}
7 1+11.3iT−49T2 1 + 11.3iT - 49T^{2}
11 1+5.87iT−121T2 1 + 5.87iT - 121T^{2}
13 1+13.9T+169T2 1 + 13.9T + 169T^{2}
17 1+11.0T+289T2 1 + 11.0T + 289T^{2}
19 1−9.34iT−361T2 1 - 9.34iT - 361T^{2}
23 1−30.4iT−529T2 1 - 30.4iT - 529T^{2}
29 1+45.5T+841T2 1 + 45.5T + 841T^{2}
31 1+49.5iT−961T2 1 + 49.5iT - 961T^{2}
37 1−48.9T+1.36e3T2 1 - 48.9T + 1.36e3T^{2}
41 1+14.8T+1.68e3T2 1 + 14.8T + 1.68e3T^{2}
43 1−5.02iT−1.84e3T2 1 - 5.02iT - 1.84e3T^{2}
47 1+83.1iT−2.20e3T2 1 + 83.1iT - 2.20e3T^{2}
53 1+53.6T+2.80e3T2 1 + 53.6T + 2.80e3T^{2}
59 1+98.3iT−3.48e3T2 1 + 98.3iT - 3.48e3T^{2}
61 1−20.4T+3.72e3T2 1 - 20.4T + 3.72e3T^{2}
67 1−17.2iT−4.48e3T2 1 - 17.2iT - 4.48e3T^{2}
71 1+52.6iT−5.04e3T2 1 + 52.6iT - 5.04e3T^{2}
73 1−98.1T+5.32e3T2 1 - 98.1T + 5.32e3T^{2}
79 1+3.04iT−6.24e3T2 1 + 3.04iT - 6.24e3T^{2}
83 1−101.iT−6.88e3T2 1 - 101. iT - 6.88e3T^{2}
89 1−17.0T+7.92e3T2 1 - 17.0T + 7.92e3T^{2}
97 1+52.0T+9.40e3T2 1 + 52.0T + 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

−10.94968329264046178591445382318, −9.824280821157405252839843939859, −9.525154730901921000888313431703, −7.945455202210446214518830502309, −7.45595270104995329475635362788, −6.41463769323066282772977344546, −5.30736832316189191765983893085, −3.85706678971392270971216109182, −1.93680067336456159358142846598, −0.32708483547886811856257630468, 2.01523218612997633520314688167, 2.77166809019414251983167633317, 4.64543502249655902662061803380, 5.97151258498658838714630486341, 7.05752672669363682068483504439, 8.183415524955897942580848246430, 9.163861386165255313695121602393, 9.589288043616126533118025612231, 10.73367044432997071754465214587, 11.69343155099150540060037288534

Graph of the ZZ-function along the critical line