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Properties

Label 2-3920-140.79-c0-0-8
Degree 22
Conductor 39203920
Sign 0.386+0.922i0.386 + 0.922i
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  + (0.5 − 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.499 − 0.866i)25-s + 2·29-s − 2·41-s + (−0.499 − 0.866i)45-s + (1 − 1.73i)61-s + (−0.499 − 0.866i)81-s + (−1 + 1.73i)89-s + (−1 − 1.73i)101-s + (1 + 1.73i)109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.499 − 0.866i)25-s + 2·29-s − 2·41-s + (−0.499 − 0.866i)45-s + (1 − 1.73i)61-s + (−0.499 − 0.866i)81-s + (−1 + 1.73i)89-s + (−1 − 1.73i)101-s + (1 + 1.73i)109-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: 0.386+0.922i0.386 + 0.922i
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), 0.386+0.922i)(2,\ 3920,\ (\ :0),\ 0.386 + 0.922i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4583379471.458337947
L(12)L(\frac12) ≈\approx 1.4583379471.458337947
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)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−T2 1 - T^{2}
17 1+(0.5−0.866i)T2 1 + (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+2T+T2 1 + 2T + 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+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-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+(0.5−0.866i)T2 1 + (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+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
97 1−T2 1 - 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

−8.544904331268213053730215793229, −7.976548706184446865390362053358, −6.78459089254537512914890994565, −6.45892390804888911059095724372, −5.43929766255881672517468222856, −4.78245619302691252748903274452, −3.99719055258620313241446925148, −3.03745872317234705303229426957, −1.85906446390295164884588257946, −0.883798214941200687861064139717, 1.48812474695699962345333133290, 2.43850170423643638790939319817, 3.19217669200412904210876527403, 4.27086280084149204760307857238, 5.06647583653431159683539426337, 5.84483935684594269645001444564, 6.75185943142755218652918484211, 7.12687349369153388765452661726, 8.076981934288995250842476762173, 8.658324089163570759377829810949

Graph of the ZZ-function along the critical line