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Properties

Label 2-3920-140.39-c0-0-8
Degree 22
Conductor 39203920
Sign −0.991+0.126i-0.991 + 0.126i
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 − 1.5i)3-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)9-s − 1.73·15-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.5 + 0.866i)61-s + (0.866 − 1.5i)67-s − 3·69-s + (0.866 + 1.49i)75-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)3-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)9-s − 1.73·15-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s − 1.73·27-s − 29-s − 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.5 + 0.866i)61-s + (0.866 − 1.5i)67-s − 3·69-s + (0.866 + 1.49i)75-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: −0.991+0.126i-0.991 + 0.126i
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), −0.991+0.126i)(2,\ 3920,\ (\ :0),\ -0.991 + 0.126i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3043214081.304321408
L(12)L(\frac12) ≈\approx 1.3043214081.304321408
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+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-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+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
29 1+T+T2 1 + T + 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+T+T2 1 + T + T^{2}
43 1−1.73T+T2 1 - 1.73T + 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)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-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 1−1.73T+T2 1 - 1.73T + T^{2}
89 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−8.079945827083462721473241028423, −7.80580038519904753069600851201, −6.97394946634529699168248529249, −6.29013004248372156181217495616, −5.42594132980299044061662632055, −4.34794937899349079919028252837, −3.55329299812998712318779805701, −2.49569765649978014150772122331, −1.73455605969730029352639096444, −0.63308463080268752261611707418, 2.07508678600864515345624840860, 2.96499161671025764846899260910, 3.74632592457904606770835707131, 4.06738419002790759168722974944, 5.14491971962024509068373537005, 5.86099364536370215759121436522, 6.96955897054420702917668266681, 7.75405688659345585798528270835, 8.255852435382614280078614128719, 9.161695875616002364993737255196

Graph of the ZZ-function along the critical line