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Properties

Label 2-2160-240.29-c0-0-4
Degree 22
Conductor 21602160
Sign 0.382+0.923i0.382 + 0.923i
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  − i·2-s − 4-s + (0.707 + 0.707i)5-s − 1.41·7-s + i·8-s + (0.707 − 0.707i)10-s + (−0.707 − 0.707i)11-s + (0.707 − 0.707i)13-s + 1.41i·14-s + 16-s + 17-s + (1 + i)19-s + (−0.707 − 0.707i)20-s + (−0.707 + 0.707i)22-s − i·23-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.707 + 0.707i)5-s − 1.41·7-s + i·8-s + (0.707 − 0.707i)10-s + (−0.707 − 0.707i)11-s + (0.707 − 0.707i)13-s + 1.41i·14-s + 16-s + 17-s + (1 + i)19-s + (−0.707 − 0.707i)20-s + (−0.707 + 0.707i)22-s − i·23-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.382+0.923i0.382 + 0.923i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(269,⋅)\chi_{2160} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), 0.382+0.923i)(2,\ 2160,\ (\ :0),\ 0.382 + 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0491777231.049177723
L(12)L(\frac12) ≈\approx 1.0491777231.049177723
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+iT 1 + iT
3 1 1
5 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good7 1+1.41T+T2 1 + 1.41T + T^{2}
11 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
13 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
17 1−T+T2 1 - T + T^{2}
19 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
23 1+iT−T2 1 + iT - T^{2}
29 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
31 1−T+T2 1 - T + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
47 1+T+T2 1 + T + T^{2}
53 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
59 1+iT2 1 + iT^{2}
61 1+iT2 1 + iT^{2}
67 1−iT2 1 - iT^{2}
71 1−1.41T+T2 1 - 1.41T + T^{2}
73 1+T2 1 + T^{2}
79 1−T+T2 1 - T + T^{2}
83 1+iT2 1 + iT^{2}
89 1+1.41T+T2 1 + 1.41T + T^{2}
97 1−1.41iT−T2 1 - 1.41iT - 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

−9.555679797801535057789549525654, −8.411899839042732464952573223106, −7.83593695529272576021714635961, −6.51365430204369063828084806564, −5.90596665185178497415205119033, −5.24465263767617297644196765222, −3.74460014154716811243422677034, −3.12546505449989127700786031103, −2.57825240248484960503014651720, −0.970317249283850218835519861589, 1.11347230893064976083318470057, 2.82505653744155938864322342252, 3.82491265488193320850354273197, 4.91850566462571358267776070422, 5.50067882348676699322112194661, 6.32492904311922242424183096741, 6.95732698762567405600377437589, 7.75792173831450288889235713108, 8.720650669136127417091031763067, 9.392340317272698266537214803785

Graph of the ZZ-function along the critical line