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Properties

Label 2-2160-15.14-c2-0-41
Degree 22
Conductor 21602160
Sign 0.852−0.522i0.852 - 0.522i
Analytic cond. 58.855758.8557
Root an. cond. 7.671747.67174
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  + (4.26 − 2.61i)5-s − 5.45i·7-s + 7.38i·11-s + 15.9i·13-s + 15.3·17-s − 2.76·19-s + 3.49·23-s + (11.3 − 22.2i)25-s + 43.6i·29-s − 1.13·31-s + (−14.2 − 23.2i)35-s + 47.5i·37-s + 6.64i·41-s + 23.1i·43-s − 39.6·47-s + ⋯
L(s)  = 1  + (0.852 − 0.522i)5-s − 0.779i·7-s + 0.670i·11-s + 1.23i·13-s + 0.900·17-s − 0.145·19-s + 0.151·23-s + (0.453 − 0.891i)25-s + 1.50i·29-s − 0.0365·31-s + (−0.407 − 0.664i)35-s + 1.28i·37-s + 0.162i·41-s + 0.539i·43-s − 0.844·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.852−0.522i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((0.852−0.522i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.852−0.522i0.852 - 0.522i
Analytic conductor: 58.855758.8557
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1), 0.852−0.522i)(2,\ 2160,\ (\ :1),\ 0.852 - 0.522i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.3940923752.394092375
L(12)L(\frac12) ≈\approx 2.3940923752.394092375
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
3 1 1
5 1+(−4.26+2.61i)T 1 + (-4.26 + 2.61i)T
good7 1+5.45iT−49T2 1 + 5.45iT - 49T^{2}
11 1−7.38iT−121T2 1 - 7.38iT - 121T^{2}
13 1−15.9iT−169T2 1 - 15.9iT - 169T^{2}
17 1−15.3T+289T2 1 - 15.3T + 289T^{2}
19 1+2.76T+361T2 1 + 2.76T + 361T^{2}
23 1−3.49T+529T2 1 - 3.49T + 529T^{2}
29 1−43.6iT−841T2 1 - 43.6iT - 841T^{2}
31 1+1.13T+961T2 1 + 1.13T + 961T^{2}
37 1−47.5iT−1.36e3T2 1 - 47.5iT - 1.36e3T^{2}
41 1−6.64iT−1.68e3T2 1 - 6.64iT - 1.68e3T^{2}
43 1−23.1iT−1.84e3T2 1 - 23.1iT - 1.84e3T^{2}
47 1+39.6T+2.20e3T2 1 + 39.6T + 2.20e3T^{2}
53 1−52.1T+2.80e3T2 1 - 52.1T + 2.80e3T^{2}
59 1−49.5iT−3.48e3T2 1 - 49.5iT - 3.48e3T^{2}
61 1+39.7T+3.72e3T2 1 + 39.7T + 3.72e3T^{2}
67 1−71.0iT−4.48e3T2 1 - 71.0iT - 4.48e3T^{2}
71 1+85.3iT−5.04e3T2 1 + 85.3iT - 5.04e3T^{2}
73 1−28.9iT−5.32e3T2 1 - 28.9iT - 5.32e3T^{2}
79 1−90.7T+6.24e3T2 1 - 90.7T + 6.24e3T^{2}
83 1+97.6T+6.88e3T2 1 + 97.6T + 6.88e3T^{2}
89 1−121.iT−7.92e3T2 1 - 121. iT - 7.92e3T^{2}
97 1+77.9iT−9.40e3T2 1 + 77.9iT - 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

−9.036544475211408717170809699318, −8.295614401106144430124580457806, −7.23284093778763854323291753290, −6.73969529694860529884349136775, −5.78653984033138311192629281259, −4.86518756901221003654950662431, −4.27230391236528859812479265031, −3.10882312592180583334986222989, −1.84605986992610680855436365983, −1.11160069731174358898775374359, 0.64729526827256238641799407595, 2.06060749311273513694059807903, 2.86888498102608667028837144881, 3.67824895067183570530447981452, 5.13409825174955149010391603215, 5.75722889788412244984041538396, 6.19867094093026955895852058458, 7.32836745476551530406538733576, 8.078195247171008443799194582110, 8.859541455912937921542051911372

Graph of the ZZ-function along the critical line