↑

Properties

Label 2-3800-152.139-c0-0-4
Degree 22
Conductor 38003800
Sign 0.631+0.775i0.631 + 0.775i
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
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  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.300 − 0.173i)7-s + (−0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.766 − 1.32i)11-s + (−0.342 − 0.939i)13-s + (−0.0603 − 0.342i)14-s + (−0.939 − 0.342i)16-s − i·18-s + (−0.766 + 0.642i)19-s + (0.524 − 1.43i)22-s + (1.85 + 0.326i)23-s + (0.5 − 0.866i)26-s + (0.223 − 0.266i)28-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.300 − 0.173i)7-s + (−0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.766 − 1.32i)11-s + (−0.342 − 0.939i)13-s + (−0.0603 − 0.342i)14-s + (−0.939 − 0.342i)16-s − i·18-s + (−0.766 + 0.642i)19-s + (0.524 − 1.43i)22-s + (1.85 + 0.326i)23-s + (0.5 − 0.866i)26-s + (0.223 − 0.266i)28-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.631+0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((0.631+0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 0.631+0.775i0.631 + 0.775i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1051,⋅)\chi_{3800} (1051, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 0.631+0.775i)(2,\ 3800,\ (\ :0),\ 0.631 + 0.775i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.93778560440.9377856044
L(12)L(\frac12) ≈\approx 0.93778560440.9377856044
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+(−0.642−0.766i)T 1 + (-0.642 - 0.766i)T
5 1 1
19 1+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
good3 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
7 1+(0.300+0.173i)T+(0.5+0.866i)T2 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2}
11 1+(0.766+1.32i)T+(−0.5+0.866i)T2 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.342+0.939i)T+(−0.766+0.642i)T2 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2}
17 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
23 1+(−1.85−0.326i)T+(0.939+0.342i)T2 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2}
29 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+1.87iT−T2 1 + 1.87iT - T^{2}
41 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
43 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
47 1+(−0.642+0.766i)T+(−0.173−0.984i)T2 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2}
53 1+(0.342+0.0603i)T+(0.939+0.342i)T2 1 + (0.342 + 0.0603i)T + (0.939 + 0.342i)T^{2}
59 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
61 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
67 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
71 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
73 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
79 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
83 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
89 1+(1.76−0.642i)T+(0.766−0.642i)T2 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2}
97 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)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

−8.582147509076677513358079958195, −7.78178078692251458932098166135, −7.00190507726258487576797954224, −6.27023452065248572059846487575, −5.46224495886188465461611485333, −5.22519275327084068249784743296, −3.79298212839026636309648412943, −3.30388673604034837825825341271, −2.54552651539510379931434189431, −0.40174486317415070186052478663, 1.60945670089102455026612486123, 2.57739145307573726846400788545, 3.01065276980924183137763759822, 4.49312287776269092158751599470, 4.75206403663292254351588920914, 5.52131506483257804116002297064, 6.61381911442416819447268218600, 7.02104523859212297588926483856, 8.178732855035040725091151812833, 8.967879113576591457841747486419

Graph of the ZZ-function along the critical line