↑

Properties

Label 2-3264-8.5-c1-0-21
Degree 22
Conductor 32643264
Sign −0.258−0.965i-0.258 - 0.965i
Analytic cond. 26.063126.0631
Root an. cond. 5.105215.10521
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 0.792i·5-s + 3.46·7-s − 9-s + 3.37i·11-s + 5.84i·13-s + 0.792·15-s − 17-s + 0.627i·19-s + 3.46i·21-s + 5.84·23-s + 4.37·25-s − i·27-s + 5.04i·29-s − 6.63·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.354i·5-s + 1.30·7-s − 0.333·9-s + 1.01i·11-s + 1.61i·13-s + 0.204·15-s − 0.242·17-s + 0.144i·19-s + 0.755i·21-s + 1.21·23-s + 0.874·25-s − 0.192i·27-s + 0.937i·29-s − 1.19·31-s + ⋯

Functional equation

Λ(s)=(3264s/2 ΓC(s) L(s)=((−0.258−0.965i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3264s/2 ΓC(s+1/2) L(s)=((−0.258−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32643264    =    26⋅3⋅172^{6} \cdot 3 \cdot 17
Sign: −0.258−0.965i-0.258 - 0.965i
Analytic conductor: 26.063126.0631
Root analytic conductor: 5.105215.10521
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3264(1633,⋅)\chi_{3264} (1633, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3264, ( :1/2), −0.258−0.965i)(2,\ 3264,\ (\ :1/2),\ -0.258 - 0.965i)

Particular Values

L(1)L(1) ≈\approx 1.9272010281.927201028
L(12)L(\frac12) ≈\approx 1.9272010281.927201028
L(32)L(\frac{3}{2}) 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−iT 1 - iT
17 1+T 1 + T
good5 1+0.792iT−5T2 1 + 0.792iT - 5T^{2}
7 1−3.46T+7T2 1 - 3.46T + 7T^{2}
11 1−3.37iT−11T2 1 - 3.37iT - 11T^{2}
13 1−5.84iT−13T2 1 - 5.84iT - 13T^{2}
19 1−0.627iT−19T2 1 - 0.627iT - 19T^{2}
23 1−5.84T+23T2 1 - 5.84T + 23T^{2}
29 1−5.04iT−29T2 1 - 5.04iT - 29T^{2}
31 1+6.63T+31T2 1 + 6.63T + 31T^{2}
37 1+3.16iT−37T2 1 + 3.16iT - 37T^{2}
41 1+8.11T+41T2 1 + 8.11T + 41T^{2}
43 1+6.11iT−43T2 1 + 6.11iT - 43T^{2}
47 1+6.92T+47T2 1 + 6.92T + 47T^{2}
53 1−1.87iT−53T2 1 - 1.87iT - 53T^{2}
59 1−2.74iT−59T2 1 - 2.74iT - 59T^{2}
61 1−5.34iT−61T2 1 - 5.34iT - 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+8.51T+71T2 1 + 8.51T + 71T^{2}
73 1−16.7T+73T2 1 - 16.7T + 73T^{2}
79 1+3.46T+79T2 1 + 3.46T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+15.4T+89T2 1 + 15.4T + 89T^{2}
97 1−14T+97T2 1 - 14T + 97T^{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

−8.902307951213832504587792682785, −8.357202769973474900036256022176, −7.16820779226728103351489670074, −6.93224184408061289520490977152, −5.55673103050149638330476316775, −4.78035233151220408613878622874, −4.53699142426775996245585584455, −3.49883433206534173990013645525, −2.12873781690962140181878015216, −1.44227188342826731851423699486, 0.60242863589734796587190692136, 1.62873536493018476074239795153, 2.81693907621653321645051906960, 3.43240745386328841798426135708, 4.84161548867832528077434306728, 5.32128684851171332735802218526, 6.19754885479902958820350685662, 6.99962239825969591739159701443, 7.85613167089990326054598526043, 8.243642599403387595021653160397

Graph of the ZZ-function along the critical line