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Properties

Label 2-3920-140.39-c0-0-6
Degree 22
Conductor 39203920
Sign 0.922+0.386i0.922 + 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+0.386i0.922 + 0.386i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(1439,⋅)\chi_{3920} (1439, \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 1.6550042451.655004245
L(12)L(\frac12) ≈\approx 1.6550042451.655004245
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

−8.418574673834591410736381274313, −7.917996879118069482524640694649, −7.36039972503599067515737213706, −6.13807167678287230776968203034, −5.44023125587389568714824232883, −5.23526744417419994762963215355, −3.96959647685308557387790409369, −3.10337384369007198507850649765, −2.03367807888651183556892160943, −1.11831237650297407011394469092, 1.33107996145596575114132561806, 2.16833957810339294480032665375, 3.30912613006016697680267459622, 3.98058925832195806117775795481, 5.04314864506762778018709152836, 5.79345676558268074906752727867, 6.58330617910361011557887571935, 7.08807659222138977817010751552, 7.76691092711698886883391325341, 9.071298528772238584678727792450

Graph of the ZZ-function along the critical line