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Properties

Label 2-2160-240.203-c0-0-0
Degree 22
Conductor 21602160
Sign 0.160−0.987i0.160 - 0.987i
Analytic cond. 1.077981.07798
Root an. cond. 1.038251.03825
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)8-s − 1.00i·10-s + (−0.707 − 0.707i)11-s + 13-s − 1.00·16-s + (0.707 + 0.707i)17-s + (0.707 + 0.707i)20-s + 1.00·22-s + (0.707 − 0.707i)23-s − 1.00i·25-s + (−0.707 + 0.707i)26-s + (−0.707 + 0.707i)29-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)8-s − 1.00i·10-s + (−0.707 − 0.707i)11-s + 13-s − 1.00·16-s + (0.707 + 0.707i)17-s + (0.707 + 0.707i)20-s + 1.00·22-s + (0.707 − 0.707i)23-s − 1.00i·25-s + (−0.707 + 0.707i)26-s + (−0.707 + 0.707i)29-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.160−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((0.160−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.160−0.987i0.160 - 0.987i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(1403,⋅)\chi_{2160} (1403, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), 0.160−0.987i)(2,\ 2160,\ (\ :0),\ 0.160 - 0.987i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.70189351940.7018935194
L(12)L(\frac12) ≈\approx 0.70189351940.7018935194
L(1)L(1) 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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1 1
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good7 1−iT2 1 - iT^{2}
11 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
13 1−T+T2 1 - T + T^{2}
17 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
19 1−iT2 1 - iT^{2}
23 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
29 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
31 1−iT−T2 1 - iT - T^{2}
37 1+T2 1 + T^{2}
41 1−1.41T+T2 1 - 1.41T + T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
53 1−1.41T+T2 1 - 1.41T + T^{2}
59 1−iT2 1 - iT^{2}
61 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
67 1+2T+T2 1 + 2T + T^{2}
71 1−T2 1 - T^{2}
73 1−iT2 1 - iT^{2}
79 1−T+T2 1 - T + T^{2}
83 1−1.41T+T2 1 - 1.41T + T^{2}
89 1−T2 1 - T^{2}
97 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{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

−9.109101892281609351663237397370, −8.629834381163140730910205372796, −7.77432679554718394243505944495, −7.36610546865132212138019818895, −6.28804878744290457656649636560, −5.86014901763654521353025591785, −4.75940237151608450439752727147, −3.66466909963963894600455414189, −2.69936718627400046150890702690, −1.11715492894546494497931521665, 0.78519822039765553627000189864, 2.03708726662288201654305360948, 3.22721566942045914690209991482, 4.02217649545060334502972794743, 4.87647789558870307351524115810, 5.87005217889429243749792127137, 7.28861518056447071799652919132, 7.62929089082333278809450056496, 8.377059729341352950321407989913, 9.216113678600207724271230018648

Graph of the ZZ-function along the critical line