↑

Properties

Label 2-800-100.11-c0-0-0
Degree 22
Conductor 800800
Sign 0.612−0.790i0.612 - 0.790i
Analytic cond. 0.3992520.399252
Root an. cond. 0.6318630.631863
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  + (−0.951 − 0.309i)3-s + (−0.809 − 0.587i)5-s + 1.61i·7-s + (1.30 + 0.951i)13-s + (0.587 + 0.809i)15-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (0.587 + 0.809i)27-s + (−0.190 + 0.587i)29-s + (0.951 − 0.309i)31-s + (0.951 − 1.30i)35-s + (0.809 + 0.587i)37-s + (−0.951 − 1.30i)39-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + (−0.809 − 0.587i)5-s + 1.61i·7-s + (1.30 + 0.951i)13-s + (0.587 + 0.809i)15-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (0.587 + 0.809i)27-s + (−0.190 + 0.587i)29-s + (0.951 − 0.309i)31-s + (0.951 − 1.30i)35-s + (0.809 + 0.587i)37-s + (−0.951 − 1.30i)39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((0.612−0.790i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s) L(s)=((0.612−0.790i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: 0.612−0.790i0.612 - 0.790i
Analytic conductor: 0.3992520.399252
Root analytic conductor: 0.6318630.631863
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ800(511,⋅)\chi_{800} (511, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :0), 0.612−0.790i)(2,\ 800,\ (\ :0),\ 0.612 - 0.790i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.56410076370.5641007637
L(12)L(\frac12) ≈\approx 0.56410076370.5641007637
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+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
good3 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
7 1−1.61iT−T2 1 - 1.61iT - T^{2}
11 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
13 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
17 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
19 1+(0.587−0.190i)T+(0.809−0.587i)T2 1 + (0.587 - 0.190i)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+(0.190−0.587i)T+(−0.809−0.587i)T2 1 + (0.190 - 0.587i)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+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
41 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
43 1+0.618iT−T2 1 + 0.618iT - T^{2}
47 1+(1.53+0.5i)T+(0.809+0.587i)T2 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2}
53 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
59 1+(−0.363+0.5i)T+(−0.309−0.951i)T2 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}
61 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
67 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
71 1+(−1.53−0.5i)T+(0.809+0.587i)T2 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2}
73 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
79 1+(0.587+0.190i)T+(0.809+0.587i)T2 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2}
83 1+(0.951−0.309i)T+(0.809−0.587i)T2 1 + (0.951 - 0.309i)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

−11.13235746859653710027954389613, −9.568961502346711834596502097167, −8.717981016387441683898159566088, −8.333387192106487678352513901161, −6.93563737475737013129668415588, −6.09735624353967895357515333516, −5.44528026384403134791621532595, −4.43959128520167464188661590950, −3.16359254788662527190539733778, −1.50269268871359637148224907544, 0.72410527832944070341814738241, 3.07466700820151543754262244854, 4.07085834810941884673309395035, 4.79450702754302185612680932296, 6.16339941026947681820757163648, 6.70994498706297363980216116141, 7.82178584472071459942098804806, 8.359431168317510759850937031981, 9.916495128276396188435742587749, 10.76421290348671592221019667730

Graph of the ZZ-function along the critical line