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Properties

Label 2-2883-93.35-c0-0-3
Degree 22
Conductor 28832883
Sign 0.0752−0.997i0.0752 - 0.997i
Analytic cond. 1.438801.43880
Root an. cond. 1.199501.19950
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  + (0.951 − 0.309i)2-s + (0.453 + 0.891i)3-s + i·5-s + (0.707 + 0.707i)6-s + (0.809 − 0.587i)7-s + (−0.587 + 0.809i)8-s + (−0.587 + 0.809i)9-s + (0.309 + 0.951i)10-s + (0.587 − 0.809i)14-s + (−0.891 + 0.453i)15-s + (−0.309 + 0.951i)16-s + (0.831 − 1.14i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.891 + 0.453i)21-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.453 + 0.891i)3-s + i·5-s + (0.707 + 0.707i)6-s + (0.809 − 0.587i)7-s + (−0.587 + 0.809i)8-s + (−0.587 + 0.809i)9-s + (0.309 + 0.951i)10-s + (0.587 − 0.809i)14-s + (−0.891 + 0.453i)15-s + (−0.309 + 0.951i)16-s + (0.831 − 1.14i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.891 + 0.453i)21-s + ⋯

Functional equation

Λ(s)=(2883s/2 ΓC(s) L(s)=((0.0752−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2883s/2 ΓC(s) L(s)=((0.0752−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28832883    =    3⋅3123 \cdot 31^{2}
Sign: 0.0752−0.997i0.0752 - 0.997i
Analytic conductor: 1.438801.43880
Root analytic conductor: 1.199501.19950
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2883(2453,⋅)\chi_{2883} (2453, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2883, ( :0), 0.0752−0.997i)(2,\ 2883,\ (\ :0),\ 0.0752 - 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1981969722.198196972
L(12)L(\frac12) ≈\approx 2.1981969722.198196972
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+(−0.453−0.891i)T 1 + (-0.453 - 0.891i)T
31 1 1
good2 1+(−0.951+0.309i)T+(0.809−0.587i)T2 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}
5 1−iT−T2 1 - iT - T^{2}
7 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
11 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
13 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
17 1+(−0.831+1.14i)T+(−0.309−0.951i)T2 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2}
19 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
23 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
29 1+(1.34−0.437i)T+(0.809−0.587i)T2 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2}
37 1+T2 1 + T^{2}
41 1+(0.951−0.309i)T+(0.809−0.587i)T2 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2}
43 1+(0.437+1.34i)T+(−0.809+0.587i)T2 1 + (0.437 + 1.34i)T + (-0.809 + 0.587i)T^{2}
47 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
53 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
59 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
61 1−1.41T+T2 1 - 1.41T + T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
73 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(1.34−0.437i)T+(0.809−0.587i)T2 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2}
89 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
97 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)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

−9.205182876801631863322374053779, −8.333772572032062358581153141962, −7.67866285491807076170336612257, −6.92437892693810631146637404501, −5.57157742432291243374935901767, −5.20272005901193215366972590022, −4.26040985071515780223231614058, −3.54584083045496564730379019852, −3.03625784569626993908862257455, −1.98250557323724143719386622107, 1.03500000886511421970374054109, 2.04177907635318126066102834741, 3.26812180278011093091308842808, 4.12710642793995584595047045811, 5.12267812142190632907088364196, 5.51013577304456418238162499889, 6.33819194192278041554242564735, 7.19572798248096167243057571377, 8.112629270844891560316066526071, 8.595108785079460221605944326031

Graph of the ZZ-function along the critical line