↑

Properties

Label 2-2160-15.14-c0-0-0
Degree 22
Conductor 21602160
Sign −0.707−0.707i-0.707 - 0.707i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)5-s + i·7-s + 1.41i·11-s − i·13-s − 19-s − 1.41·23-s − 1.00i·25-s + 1.41i·29-s + (−0.707 − 0.707i)35-s + i·37-s − 1.41i·41-s − 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (0.707 + 0.707i)65-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s + i·7-s + 1.41i·11-s − i·13-s − 19-s − 1.41·23-s − 1.00i·25-s + 1.41i·29-s + (−0.707 − 0.707i)35-s + i·37-s − 1.41i·41-s − 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (0.707 + 0.707i)65-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), −0.707−0.707i)(2,\ 2160,\ (\ :0),\ -0.707 - 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69033791100.6903379110
L(12)L(\frac12) ≈\approx 0.69033791100.6903379110
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 1
3 1 1
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good7 1−iT−T2 1 - iT - T^{2}
11 1−1.41iT−T2 1 - 1.41iT - T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+T2 1 + T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+1.41T+T2 1 + 1.41T + T^{2}
29 1−1.41iT−T2 1 - 1.41iT - T^{2}
31 1+T2 1 + T^{2}
37 1−iT−T2 1 - iT - T^{2}
41 1+1.41iT−T2 1 + 1.41iT - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1+1.41T+T2 1 + 1.41T + T^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1−1.41iT−T2 1 - 1.41iT - T^{2}
73 1−iT−T2 1 - iT - T^{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−iT−T2 1 - iT - T^{2}
show more
show less
   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

−9.619713358843898470652388011208, −8.626772884246181481728064390979, −8.033633710346540186562753662248, −7.24232143035549380544886447272, −6.51238460270033905146623495305, −5.63076908321124070471609486504, −4.71388620877052884621676361421, −3.81147327691642548807846911105, −2.79236654733543617383755313581, −1.95934878714665840016153863370, 0.46525003722821184951591825544, 1.86200313568443260690063170890, 3.39341794277616475597240913041, 4.12912090296278936926376274417, 4.67362916347042190253038986671, 5.99004745699615919943991739275, 6.50694258561531982445029001036, 7.77110295266367842309127854352, 8.002421206310358632771626564699, 8.944814900947348085536654916066

Graph of the ZZ-function along the critical line