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Properties

Label 2-3920-140.79-c0-0-9
Degree 22
Conductor 39203920
Sign −0.922+0.386i-0.922 + 0.386i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)5-s + (0.5 − 0.866i)9-s − 2i·13-s + (−1.73 + i)17-s + (0.499 + 0.866i)25-s − 2·29-s + (−0.866 + 0.499i)45-s + (−1 + 1.73i)65-s + (−1.73 + i)73-s + (−0.499 − 0.866i)81-s + 1.99·85-s − 2i·97-s + (−1 − 1.73i)109-s + (−1.73 − i)117-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s + (0.5 − 0.866i)9-s − 2i·13-s + (−1.73 + i)17-s + (0.499 + 0.866i)25-s − 2·29-s + (−0.866 + 0.499i)45-s + (−1 + 1.73i)65-s + (−1.73 + i)73-s + (−0.499 − 0.866i)81-s + 1.99·85-s − 2i·97-s + (−1 − 1.73i)109-s + (−1.73 − i)117-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.922+0.386i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.922+0.386i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: −0.922+0.386i-0.922 + 0.386i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(79,⋅)\chi_{3920} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3920, ( :0), −0.922+0.386i)(2,\ 3920,\ (\ :0),\ -0.922 + 0.386i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52488495860.5248849586
L(12)L(\frac12) ≈\approx 0.52488495860.5248849586
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
7 1 1
good3 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+2iT−T2 1 + 2iT - T^{2}
17 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+2T+T2 1 + 2T + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+2iT−T2 1 + 2iT - T^{2}
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   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

−8.378293272833232963978055866766, −7.61581453727304512655077896812, −7.00501857279393429120269258992, −6.04344752980260771131176227772, −5.37285176903902925405164078577, −4.34068765620422180521038197507, −3.80565058961165716497140631245, −2.95879098209780670412631301507, −1.59431164987275454556467023760, −0.28549954460240961735295879634, 1.81962186789723008512524606350, 2.50630904097057243613620254514, 3.78714565461400642748245527060, 4.38458918026463373274329170753, 4.96227370926182661166792839586, 6.22215907784131795899957313471, 7.06740128374258918493847109610, 7.22820608677364547537620961955, 8.166481060483467310752522826861, 9.089526576797051116936218407792

Graph of the ZZ-function along the critical line