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Properties

Label 2-825-825.527-c0-0-1
Degree 22
Conductor 825825
Sign −0.992+0.125i-0.992 + 0.125i
Analytic cond. 0.4117280.411728
Root an. cond. 0.6416600.641660
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  + (−0.309 − 0.951i)3-s + (−0.951 − 0.309i)4-s − i·5-s + (−0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + 0.999i·12-s + (−0.951 + 0.309i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)20-s + (−1.76 − 0.278i)23-s − 25-s + (0.809 + 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.221 − 1.39i)37-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.951 − 0.309i)4-s − i·5-s + (−0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + 0.999i·12-s + (−0.951 + 0.309i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)20-s + (−1.76 − 0.278i)23-s − 25-s + (0.809 + 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.221 − 1.39i)37-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((−0.992+0.125i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s) L(s)=((−0.992+0.125i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −0.992+0.125i-0.992 + 0.125i
Analytic conductor: 0.4117280.411728
Root analytic conductor: 0.6416600.641660
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ825(527,⋅)\chi_{825} (527, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :0), −0.992+0.125i)(2,\ 825,\ (\ :0),\ -0.992 + 0.125i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.47951754740.4795175474
L(12)L(\frac12) ≈\approx 0.47951754740.4795175474
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)
bad3 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
5 1+iT 1 + iT
11 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
good2 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
7 1−iT2 1 - iT^{2}
13 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
17 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
19 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
23 1+(1.76+0.278i)T+(0.951+0.309i)T2 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2}
29 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
31 1+(−0.363−1.11i)T+(−0.809+0.587i)T2 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2}
37 1+(0.221+1.39i)T+(−0.951+0.309i)T2 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2}
41 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(0.642+1.26i)T+(−0.587+0.809i)T2 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2}
53 1+(−1.76+0.896i)T+(0.587−0.809i)T2 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2}
59 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
61 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
67 1+(−0.809+1.58i)T+(−0.587−0.809i)T2 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2}
71 1+(−1.11−0.363i)T+(0.809+0.587i)T2 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2}
73 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
79 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
83 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
89 1+(−1.53+1.11i)T+(0.309−0.951i)T2 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2}
97 1+(−0.278+0.142i)T+(0.587−0.809i)T2 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)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

−9.999988787068063262373544815544, −8.940677903769696787391798573652, −8.339165320279189554666847044986, −7.73943879178065716129610778529, −6.32839715444749050715654116857, −5.55822361481768071714962103257, −4.89782099441090008358061899639, −3.66551165908730171829921417021, −1.93953211410501952469502770082, −0.51276616136996274215340704142, 2.58710438200648431660924031663, 3.72749680691850918128272554673, 4.42980121086140964642187040003, 5.44547427814806156324732161567, 6.34207034023721292147924103846, 7.59709474403710844062221017793, 8.298243689854554743561949277097, 9.470342331301778581229371093891, 9.990980069323676639454475340936, 10.52920802422778068465786180664

Graph of the ZZ-function along the critical line