↑

Properties

Label 2-2160-240.149-c0-0-7
Degree 22
Conductor 21602160
Sign −0.608−0.793i-0.608 - 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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)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 − 1.93·17-s + (−0.366 + 0.366i)19-s + (0.965 + 0.258i)20-s − 1.93i·23-s + 1.00i·25-s − 1.73·31-s + (−0.965 − 0.258i)32-s + (0.499 + 1.86i)34-s + (0.448 + 0.258i)38-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)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 − 1.93·17-s + (−0.366 + 0.366i)19-s + (0.965 + 0.258i)20-s − 1.93i·23-s + 1.00i·25-s − 1.73·31-s + (−0.965 − 0.258i)32-s + (0.499 + 1.86i)34-s + (0.448 + 0.258i)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: −0.608−0.793i-0.608 - 0.793i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(1349,⋅)\chi_{2160} (1349, \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 0.18777720880.1877772088
L(12)L(\frac12) ≈\approx 0.18777720880.1877772088
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+(0.258+0.965i)T 1 + (0.258 + 0.965i)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+1.93T+T2 1 + 1.93T + T^{2}
19 1+(0.366−0.366i)T−iT2 1 + (0.366 - 0.366i)T - iT^{2}
23 1+1.93iT−T2 1 + 1.93iT - 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+(−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+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}
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

−8.687894005667826125572206753152, −8.432658945437036359953509387603, −7.42029000348678384467239691173, −6.50841802203037545644025924211, −5.22824164434273796825583794599, −4.42055326699267940563424323546, −3.93531829538049800467284191464, −2.70753123968109366687122760993, −1.70291367149656094349483665737, −0.13946743865191874362584295474, 1.92471852166543379741744696544, 3.40607197094563245814478633429, 4.18803425205742804271935829671, 5.07482965589896514407475055165, 6.02732242548462694280413575190, 6.90096662777749407372671659741, 7.24419114749213409134920593290, 8.146139713815036468085229734611, 8.843735818129565999537630856763, 9.535956104831781341166008958239

Graph of the ZZ-function along the critical line