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Properties

Label 2-3920-980.659-c0-0-0
Degree 22
Conductor 39203920
Sign −0.871−0.490i-0.871 - 0.490i
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  + (−0.433 + 1.90i)3-s + (0.222 − 0.974i)5-s + (0.781 + 0.623i)7-s + (−2.52 − 1.21i)9-s + (1.75 + 0.846i)15-s + (−1.52 + 1.21i)21-s + (−1.21 + 1.52i)23-s + (−0.900 − 0.433i)25-s + (2.19 − 2.74i)27-s + (0.777 + 0.974i)29-s + (0.781 − 0.623i)35-s + (−0.400 + 1.75i)41-s + (0.193 + 0.846i)43-s + (−1.74 + 2.19i)45-s + (−1.75 + 0.846i)47-s + ⋯
L(s)  = 1  + (−0.433 + 1.90i)3-s + (0.222 − 0.974i)5-s + (0.781 + 0.623i)7-s + (−2.52 − 1.21i)9-s + (1.75 + 0.846i)15-s + (−1.52 + 1.21i)21-s + (−1.21 + 1.52i)23-s + (−0.900 − 0.433i)25-s + (2.19 − 2.74i)27-s + (0.777 + 0.974i)29-s + (0.781 − 0.623i)35-s + (−0.400 + 1.75i)41-s + (0.193 + 0.846i)43-s + (−1.74 + 2.19i)45-s + (−1.75 + 0.846i)47-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.871−0.490i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.871−0.490i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: −0.871−0.490i-0.871 - 0.490i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(3599,⋅)\chi_{3920} (3599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3920, ( :0), −0.871−0.490i)(2,\ 3920,\ (\ :0),\ -0.871 - 0.490i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.95182086290.9518208629
L(12)L(\frac12) ≈\approx 0.95182086290.9518208629
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+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
7 1+(−0.781−0.623i)T 1 + (-0.781 - 0.623i)T
good3 1+(0.433−1.90i)T+(−0.900−0.433i)T2 1 + (0.433 - 1.90i)T + (-0.900 - 0.433i)T^{2}
11 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
13 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
17 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
19 1−T2 1 - T^{2}
23 1+(1.21−1.52i)T+(−0.222−0.974i)T2 1 + (1.21 - 1.52i)T + (-0.222 - 0.974i)T^{2}
29 1+(−0.777−0.974i)T+(−0.222+0.974i)T2 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2}
31 1−T2 1 - T^{2}
37 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
41 1+(0.400−1.75i)T+(−0.900−0.433i)T2 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2}
43 1+(−0.193−0.846i)T+(−0.900+0.433i)T2 1 + (-0.193 - 0.846i)T + (-0.900 + 0.433i)T^{2}
47 1+(1.75−0.846i)T+(0.623−0.781i)T2 1 + (1.75 - 0.846i)T + (0.623 - 0.781i)T^{2}
53 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
59 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
61 1+(−0.277−0.347i)T+(−0.222+0.974i)T2 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2}
67 1+T2 1 + T^{2}
71 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
73 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
79 1−T2 1 - T^{2}
83 1+(−1.40−0.678i)T+(0.623+0.781i)T2 1 + (-1.40 - 0.678i)T + (0.623 + 0.781i)T^{2}
89 1+(1.12+0.541i)T+(0.623+0.781i)T2 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)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

−9.125812896460376698884526672810, −8.387059753036588457746030094404, −7.928754486611322472609497400042, −6.28952557321556496398115935224, −5.72801660239782599360397062101, −4.96891028025994132551851277111, −4.69272591980936374200739468752, −3.80085033021218206614568683505, −2.90173976585988230639534834357, −1.50240872484030482504091422657, 0.56026929182649568468690408122, 1.94034876758267210907675017420, 2.28418797343952218187227656148, 3.49532987844033660693507860589, 4.71682723768485562391969072537, 5.70686668149836832176437208069, 6.32340870942257183580325787031, 6.92069140761035104750066702667, 7.42870050454100778211000839817, 8.182029069633625406756150925212

Graph of the ZZ-function along the critical line