↑

Properties

Label 2-3920-140.39-c0-0-7
Degree 22
Conductor 39203920
Sign 0.126+0.991i0.126 + 0.991i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)3-s + (−0.866 + 0.5i)5-s + (−1 − 1.73i)9-s + (1.5 + 0.866i)11-s − i·13-s + 1.73i·15-s + (0.866 + 0.5i)17-s + (0.499 − 0.866i)25-s − 1.73·27-s + 29-s + (2.59 − 1.5i)33-s + (−1.5 − 0.866i)39-s + (1.73 + i)45-s + (−0.866 − 1.5i)47-s + (1.5 − 0.866i)51-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)3-s + (−0.866 + 0.5i)5-s + (−1 − 1.73i)9-s + (1.5 + 0.866i)11-s − i·13-s + 1.73i·15-s + (0.866 + 0.5i)17-s + (0.499 − 0.866i)25-s − 1.73·27-s + 29-s + (2.59 − 1.5i)33-s + (−1.5 − 0.866i)39-s + (1.73 + i)45-s + (−0.866 − 1.5i)47-s + (1.5 − 0.866i)51-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: 0.126+0.991i0.126 + 0.991i
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), 0.126+0.991i)(2,\ 3920,\ (\ :0),\ 0.126 + 0.991i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6412919191.641291919
L(12)L(\frac12) ≈\approx 1.6412919191.641291919
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+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
11 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)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−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+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-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+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (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+iT−T2 1 + iT - T^{2}
show more
show less
   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

−8.311630600077863112621928465500, −7.71322676547030918159909370536, −7.07988781302823446588534541924, −6.63092495448071342563383752687, −5.82210702507639454374612060835, −4.48208332420018558220197383643, −3.54086961742850294143380672087, −3.00089807536480077262521595967, −1.91348192347442417615562820089, −0.995031354617777728326785980239, 1.33472995810036280332713341613, 2.93238948403121787022881030754, 3.49810450550787718276406414933, 4.27655559800699244905999742179, 4.61663693854520706083596093571, 5.65676714092821376924725208124, 6.64249872765774820952862356646, 7.59999495501815697869863599456, 8.410059671605904843041228079894, 8.863470281164058144540749964247

Graph of the ZZ-function along the critical line