↑

Properties

Label 2-3920-140.47-c0-0-3
Degree 22
Conductor 39203920
Sign −0.997+0.0674i-0.997 + 0.0674i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
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  + (0.130 − 0.991i)5-s + (−0.866 − 0.5i)9-s + (−0.541 − 0.541i)13-s + (−1.78 − 0.478i)17-s + (−0.965 − 0.258i)25-s + 1.41i·29-s + (−1.93 + 0.517i)37-s + 0.765i·41-s + (−0.608 + 0.793i)45-s + (1.36 + 0.366i)53-s + (−1.60 − 0.923i)61-s + (−0.607 + 0.465i)65-s + (0.478 − 1.78i)73-s + (0.499 + 0.866i)81-s + (−0.707 + 1.70i)85-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)5-s + (−0.866 − 0.5i)9-s + (−0.541 − 0.541i)13-s + (−1.78 − 0.478i)17-s + (−0.965 − 0.258i)25-s + 1.41i·29-s + (−1.93 + 0.517i)37-s + 0.765i·41-s + (−0.608 + 0.793i)45-s + (1.36 + 0.366i)53-s + (−1.60 − 0.923i)61-s + (−0.607 + 0.465i)65-s + (0.478 − 1.78i)73-s + (0.499 + 0.866i)81-s + (−0.707 + 1.70i)85-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.997+0.0674i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.997+0.0674i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: −0.997+0.0674i-0.997 + 0.0674i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(607,⋅)\chi_{3920} (607, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3920, ( :0), −0.997+0.0674i)(2,\ 3920,\ (\ :0),\ -0.997 + 0.0674i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.41701547860.4170154786
L(12)L(\frac12) ≈\approx 0.41701547860.4170154786
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
5 1+(−0.130+0.991i)T 1 + (-0.130 + 0.991i)T
7 1 1
good3 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.541+0.541i)T+iT2 1 + (0.541 + 0.541i)T + iT^{2}
17 1+(1.78+0.478i)T+(0.866+0.5i)T2 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
29 1−1.41iT−T2 1 - 1.41iT - T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(1.93−0.517i)T+(0.866−0.5i)T2 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2}
41 1−0.765iT−T2 1 - 0.765iT - T^{2}
43 1+iT2 1 + iT^{2}
47 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
53 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1.60+0.923i)T+(0.5+0.866i)T2 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.478+1.78i)T+(−0.866−0.5i)T2 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2}
79 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
83 1−iT2 1 - iT^{2}
89 1+(−0.923+1.60i)T+(−0.5−0.866i)T2 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2}
97 1+(0.541−0.541i)T−iT2 1 + (0.541 - 0.541i)T - iT^{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

−8.571487292221963607610546420942, −7.61153878669055602052904754317, −6.76550311167504924857608904930, −6.04901600678810999871483439901, −5.10900887955510954605990467813, −4.75422515140150648110470938914, −3.61860791508256109888457532214, −2.72200932428083174981103315663, −1.65924969479245898213628193011, −0.21094263446537228234223033387, 2.11563345474380374881956938979, 2.44066338251405531653884091878, 3.61524590466107907144262445551, 4.39753602341860972543527169078, 5.38450788781303946361307574258, 6.12351620558467718970969699773, 6.84005416794806478157653269173, 7.39485864727103542173747359851, 8.353694151212366352687423471418, 8.906191500491707193605998719443

Graph of the ZZ-function along the critical line