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Properties

Label 2-825-825.428-c0-0-1
Degree 22
Conductor 825825
Sign 0.770+0.637i0.770 + 0.637i
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  + (0.809 − 0.587i)3-s + (0.587 − 0.809i)4-s + i·5-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s − i·12-s + (0.587 + 0.809i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)20-s + (0.896 + 1.76i)23-s − 25-s + (−0.309 − 0.951i)27-s + (−1.53 + 1.11i)31-s + (−0.951 + 0.309i)33-s + (−0.587 − 0.809i)36-s + (1.26 + 0.642i)37-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.587 − 0.809i)4-s + i·5-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s − i·12-s + (0.587 + 0.809i)15-s + (−0.309 − 0.951i)16-s + (0.809 + 0.587i)20-s + (0.896 + 1.76i)23-s − 25-s + (−0.309 − 0.951i)27-s + (−1.53 + 1.11i)31-s + (−0.951 + 0.309i)33-s + (−0.587 − 0.809i)36-s + (1.26 + 0.642i)37-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((0.770+0.637i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s) L(s)=((0.770+0.637i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: 0.770+0.637i0.770 + 0.637i
Analytic conductor: 0.4117280.411728
Root analytic conductor: 0.6416600.641660
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ825(428,⋅)\chi_{825} (428, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :0), 0.770+0.637i)(2,\ 825,\ (\ :0),\ 0.770 + 0.637i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3596452361.359645236
L(12)L(\frac12) ≈\approx 1.3596452361.359645236
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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
5 1−iT 1 - iT
11 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
good2 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
7 1+iT2 1 + iT^{2}
13 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
17 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
19 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
23 1+(−0.896−1.76i)T+(−0.587+0.809i)T2 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2}
29 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
31 1+(1.53−1.11i)T+(0.309−0.951i)T2 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2}
37 1+(−1.26−0.642i)T+(0.587+0.809i)T2 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2}
41 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+(0.221+1.39i)T+(−0.951+0.309i)T2 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2}
53 1+(0.896−0.142i)T+(0.951−0.309i)T2 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2}
59 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}
61 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
67 1+(0.309−1.95i)T+(−0.951−0.309i)T2 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2}
71 1+(1.11−1.53i)T+(−0.309−0.951i)T2 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2}
73 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
89 1+(−0.363+1.11i)T+(−0.809−0.587i)T2 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2}
97 1+(−1.76+0.278i)T+(0.951−0.309i)T2 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)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

−10.26657988353649189004308478354, −9.644422959695691692352409329735, −8.606296174842911274105140839167, −7.43491212514371568072953418577, −7.14556894341914639129894191381, −6.10899466827752039359948944292, −5.25823698345561762098616928990, −3.48853972864093085692149660726, −2.72543769005277565590877891230, −1.63580873263475397377844462174, 2.08687701078055480369417199867, 3.00385941876144152960033456260, 4.21251451940156494364097363107, 4.87614385524366768937533175071, 6.15446634983255904172094144116, 7.63600956806679753812114580051, 7.84029106228471811457179564866, 8.899058665908273714747207400993, 9.397341573134603665171294555925, 10.60668951350543692884817317498

Graph of the ZZ-function along the critical line