↑

Properties

Label 2-1792-56.51-c0-0-3
Degree 22
Conductor 17921792
Sign −0.980−0.197i-0.980 - 0.197i
Analytic cond. 0.8943240.894324
Root an. cond. 0.9456870.945687
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  + (−0.5 − 0.866i)3-s + (−0.866 − 0.5i)5-s + i·7-s + (−0.5 − 0.866i)11-s + 0.999i·15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)23-s − 27-s + (−0.866 + 0.5i)31-s + (−0.499 + 0.866i)33-s + (0.5 − 0.866i)35-s + (0.866 + 0.5i)37-s + (−0.866 − 0.5i)47-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.866 − 0.5i)5-s + i·7-s + (−0.5 − 0.866i)11-s + 0.999i·15-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)23-s − 27-s + (−0.866 + 0.5i)31-s + (−0.499 + 0.866i)33-s + (0.5 − 0.866i)35-s + (0.866 + 0.5i)37-s + (−0.866 − 0.5i)47-s + ⋯

Functional equation

Λ(s)=(1792s/2 ΓC(s) L(s)=((−0.980−0.197i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1792s/2 ΓC(s) L(s)=((−0.980−0.197i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17921792    =    28⋅72^{8} \cdot 7
Sign: −0.980−0.197i-0.980 - 0.197i
Analytic conductor: 0.8943240.894324
Root analytic conductor: 0.9456870.945687
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1792(639,⋅)\chi_{1792} (639, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1792, ( :0), −0.980−0.197i)(2,\ 1792,\ (\ :0),\ -0.980 - 0.197i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.28057438320.2805743832
L(12)L(\frac12) ≈\approx 0.28057438320.2805743832
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
7 1−iT 1 - iT
good3 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
37 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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

−8.849820468524537487070727735223, −8.161231243049619037377703541598, −7.63190447264036119939575093191, −6.51850298501953270022830576239, −5.97189343762250297436720480127, −5.09712246972117181513083375882, −4.09100826103085448346625305264, −2.95914926478444190105262568794, −1.75506005225363068231710062209, −0.21465879143512795629350936273, 2.00943977961081110898327009254, 3.47624011676942146643317409881, 4.23103050687062354108591294440, 4.67657075539469320318203498803, 5.81258249262142013388132142431, 6.84673841201784765932089243989, 7.54595475639170080642710349477, 8.089498539036542899285108457135, 9.374499619947417920513617672338, 10.01101147544704990243433308913

Graph of the ZZ-function along the critical line