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Properties

Label 2-2160-240.29-c0-0-6
Degree 22
Conductor 21602160
Sign 0.991+0.130i0.991 + 0.130i
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.707 − 0.707i)8-s + (−0.500 + 0.866i)10-s + (0.500 − 0.866i)16-s + 0.517·17-s + (1.36 + 1.36i)19-s + (−0.258 + 0.965i)20-s − 0.517i·23-s − 1.00i·25-s + 1.73·31-s + (0.258 − 0.965i)32-s + (0.499 − 0.133i)34-s + (1.67 + 0.965i)38-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.707 − 0.707i)8-s + (−0.500 + 0.866i)10-s + (0.500 − 0.866i)16-s + 0.517·17-s + (1.36 + 1.36i)19-s + (−0.258 + 0.965i)20-s − 0.517i·23-s − 1.00i·25-s + 1.73·31-s + (0.258 − 0.965i)32-s + (0.499 − 0.133i)34-s + (1.67 + 0.965i)38-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.991+0.130i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((0.991+0.130i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.991+0.130i0.991 + 0.130i
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), 0.991+0.130i)(2,\ 2160,\ (\ :0),\ 0.991 + 0.130i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0205875572.020587557
L(12)L(\frac12) ≈\approx 2.0205875572.020587557
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+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
3 1 1
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good7 1+T2 1 + T^{2}
11 1+iT2 1 + iT^{2}
13 1−iT2 1 - iT^{2}
17 1−0.517T+T2 1 - 0.517T + T^{2}
19 1+(−1.36−1.36i)T+iT2 1 + (-1.36 - 1.36i)T + iT^{2}
23 1+0.517iT−T2 1 + 0.517iT - T^{2}
29 1−iT2 1 - iT^{2}
31 1−1.73T+T2 1 - 1.73T + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1+iT2 1 + iT^{2}
47 1+1.41T+T2 1 + 1.41T + T^{2}
53 1+(1.22−1.22i)T−iT2 1 + (1.22 - 1.22i)T - iT^{2}
59 1+iT2 1 + iT^{2}
61 1+(1.36+1.36i)T+iT2 1 + (1.36 + 1.36i)T + iT^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1+T+T2 1 + T + T^{2}
83 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - 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

−9.570154790350715105550016537471, −8.015506872958612472310954824371, −7.77141944380481532382919997268, −6.67599528886480564516435730746, −6.15320985926126871135944656160, −5.14136201317314403099554303208, −4.31249022713211074496259772437, −3.37765107572323775135008105252, −2.85787584510967764741605991585, −1.43077858161273097815457052874, 1.32986421537821225354457699234, 2.88293783351315656184906670919, 3.54067361044195179921401347509, 4.71770579692872513338808996835, 4.98687878794457307218893570574, 6.03400900501140486274073434248, 6.93434460139263097614863614342, 7.67960707804165621390199472094, 8.228315189664579145157785561193, 9.187682203148868289193673424541

Graph of the ZZ-function along the critical line