↑

Properties

Label 2-40e2-200.11-c0-0-2
Degree 22
Conductor 16001600
Sign 0.790+0.612i0.790 + 0.612i
Analytic cond. 0.7985040.798504
Root an. cond. 0.8935900.893590
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  + (−0.190 + 0.587i)3-s + (−0.951 − 0.309i)5-s − i·7-s + (0.5 + 0.363i)9-s + (0.809 − 0.587i)11-s + (−0.587 + 0.809i)13-s + (0.363 − 0.5i)15-s + (−0.190 − 0.587i)17-s + (−0.5 − 1.53i)19-s + (0.587 + 0.190i)21-s + (−0.587 − 0.809i)23-s + (0.809 + 0.587i)25-s + (−0.809 + 0.587i)27-s + (0.951 + 0.309i)29-s + (0.951 − 0.309i)31-s + ⋯
L(s)  = 1  + (−0.190 + 0.587i)3-s + (−0.951 − 0.309i)5-s − i·7-s + (0.5 + 0.363i)9-s + (0.809 − 0.587i)11-s + (−0.587 + 0.809i)13-s + (0.363 − 0.5i)15-s + (−0.190 − 0.587i)17-s + (−0.5 − 1.53i)19-s + (0.587 + 0.190i)21-s + (−0.587 − 0.809i)23-s + (0.809 + 0.587i)25-s + (−0.809 + 0.587i)27-s + (0.951 + 0.309i)29-s + (0.951 − 0.309i)31-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.790+0.612i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((0.790+0.612i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.790+0.612i0.790 + 0.612i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(1311,⋅)\chi_{1600} (1311, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), 0.790+0.612i)(2,\ 1600,\ (\ :0),\ 0.790 + 0.612i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87909737030.8790973703
L(12)L(\frac12) ≈\approx 0.87909737030.8790973703
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+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
good3 1+(0.190−0.587i)T+(−0.809−0.587i)T2 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}
7 1+iT−T2 1 + iT - T^{2}
11 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
13 1+(0.587−0.809i)T+(−0.309−0.951i)T2 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2}
17 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}
19 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}
23 1+(0.587+0.809i)T+(−0.309+0.951i)T2 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2}
29 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
31 1+(−0.951+0.309i)T+(0.809−0.587i)T2 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}
37 1+(−0.951+1.30i)T+(−0.309−0.951i)T2 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2}
41 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
53 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
59 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
61 1+(−0.363−0.5i)T+(−0.309+0.951i)T2 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2}
67 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
71 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
73 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
79 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
83 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
89 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
97 1+(0.5−1.53i)T+(−0.809−0.587i)T2 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)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

−9.384386592226789053726971748527, −8.906248548653584937728424893625, −7.84710650641998997001473933154, −7.11474946581388964399105293778, −6.52347748400643249944356308688, −5.05189164040826793547765312932, −4.22332973359119128743363745230, −4.10406886351164268100080291185, −2.57799399665655814024056689356, −0.798171941341680296420044569896, 1.41812033966594888991899230291, 2.65812602483812358060101301225, 3.80547869412704237945039475921, 4.56495705478741003235026362961, 5.88710288014835531906135789299, 6.42956243227564473999446142045, 7.34599149209054378889615897648, 8.036949970493345011302227447105, 8.663252638993537674353055199584, 9.872419593208841336675240633690

Graph of the ZZ-function along the critical line