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Properties

Label 2-2160-80.19-c0-0-0
Degree 22
Conductor 21602160
Sign 0.608−0.793i0.608 - 0.793i
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 + 1.93i·17-s + (−1.36 + 1.36i)19-s + (−0.258 − 0.965i)20-s + 0.517i·23-s + 1.00i·25-s − i·31-s + (−0.258 − 0.965i)32-s + (0.499 − 1.86i)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 + 1.93i·17-s + (−1.36 + 1.36i)19-s + (−0.258 − 0.965i)20-s + 0.517i·23-s + 1.00i·25-s − i·31-s + (−0.258 − 0.965i)32-s + (0.499 − 1.86i)34-s + (1.67 − 0.965i)38-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.608−0.793i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((0.608−0.793i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.608−0.793i0.608 - 0.793i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(1459,⋅)\chi_{2160} (1459, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), 0.608−0.793i)(2,\ 2160,\ (\ :0),\ 0.608 - 0.793i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50737069610.5073706961
L(12)L(\frac12) ≈\approx 0.50737069610.5073706961
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 1−1.93iT−T2 1 - 1.93iT - 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+iT−T2 1 + iT - 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+(0.366+0.366i)T+iT2 1 + (0.366 + 0.366i)T + iT^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1−1.73iT−T2 1 - 1.73iT - T^{2}
83 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)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

−9.173162764443802924345086097843, −8.583625704647858337024889065341, −8.016652949355654006477835670804, −7.41839880940624150507255602132, −6.27665574855962067544408979897, −5.67743619891521933673443141791, −4.05789509306190249838425002551, −3.83523755571541340468162986534, −2.28136923387180528574986292052, −1.28640449778082415239510948007, 0.51826993360904595937007089043, 2.35567183302053413406086109627, 2.96377883081000609096547569858, 4.32379546558310405755552285077, 5.26117657174370886429455157181, 6.40875171508268666782621023849, 7.06965871587482881501547023200, 7.41361849347263291077823645139, 8.567253422774888853841064442608, 8.910676713099752165168887931268

Graph of the ZZ-function along the critical line