↑

Properties

Label 2-3920-140.87-c0-0-3
Degree 22
Conductor 39203920
Sign 0.284+0.958i0.284 + 0.958i
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  + (−0.608 + 0.793i)5-s + (0.866 − 0.5i)9-s + (−1.30 − 1.30i)13-s + (−0.198 − 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s − 1.84i·41-s + (−0.130 + 0.991i)45-s + (−0.366 − 1.36i)53-s + (0.662 − 0.382i)61-s + (1.83 − 0.241i)65-s + (0.739 − 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)5-s + (0.866 − 0.5i)9-s + (−1.30 − 1.30i)13-s + (−0.198 − 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s − 1.84i·41-s + (−0.130 + 0.991i)45-s + (−0.366 − 1.36i)53-s + (0.662 − 0.382i)61-s + (1.83 − 0.241i)65-s + (0.739 − 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((0.284+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((0.284+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: 0.284+0.958i0.284 + 0.958i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(3167,⋅)\chi_{3920} (3167, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3920, ( :0), 0.284+0.958i)(2,\ 3920,\ (\ :0),\ 0.284 + 0.958i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.89430591350.8943059135
L(12)L(\frac12) ≈\approx 0.89430591350.8943059135
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+(0.608−0.793i)T 1 + (0.608 - 0.793i)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+(1.30+1.30i)T+iT2 1 + (1.30 + 1.30i)T + iT^{2}
17 1+(0.198+0.739i)T+(−0.866+0.5i)T2 1 + (0.198 + 0.739i)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+(0.517−1.93i)T+(−0.866−0.5i)T2 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2}
41 1+1.84iT−T2 1 + 1.84iT - T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
53 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.662+0.382i)T+(0.5−0.866i)T2 1 + (-0.662 + 0.382i)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+(−0.739+0.198i)T+(0.866−0.5i)T2 1 + (-0.739 + 0.198i)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+(−0.382−0.662i)T+(−0.5+0.866i)T2 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2}
97 1+(1.30−1.30i)T−iT2 1 + (1.30 - 1.30i)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

−8.233967775076246375225354281640, −7.74693397520141636689738655914, −7.00894108174566357479730724909, −6.56416632512933884838246848902, −5.43768150073276131337388792568, −4.69689093522186642050992803082, −3.80616023736907772205232464956, −3.02837965193462059062018397428, −2.19674684513873403396136706128, −0.51374411589060138566693037368, 1.39682455406892899880959016098, 2.22473271480746796021135779839, 3.56617801428978443922683506468, 4.46405601707140602367830253376, 4.74938273512365047621257575672, 5.72146129057019962580539456823, 6.85996596192851180200347819017, 7.31292841051026243875434092878, 8.004022700590438463929411999559, 8.868038708709022086321428561936

Graph of the ZZ-function along the critical line