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Properties

Label 2-2793-2793.1472-c0-0-0
Degree 22
Conductor 27932793
Sign 0.999−0.00868i0.999 - 0.00868i
Analytic cond. 1.393881.39388
Root an. cond. 1.180631.18063
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  + (−0.318 − 0.947i)3-s + (−0.969 − 0.246i)4-s + (−0.998 + 0.0498i)7-s + (−0.797 + 0.603i)9-s + (0.0747 + 0.997i)12-s + (−0.603 + 1.17i)13-s + (0.878 + 0.478i)16-s + (−0.222 − 0.974i)19-s + (0.365 + 0.930i)21-s + (0.921 + 0.388i)25-s + (0.826 + 0.563i)27-s + (0.980 + 0.198i)28-s + 1.39·31-s + (0.921 − 0.388i)36-s + (−1.60 + 1.09i)37-s + ⋯
L(s)  = 1  + (−0.318 − 0.947i)3-s + (−0.969 − 0.246i)4-s + (−0.998 + 0.0498i)7-s + (−0.797 + 0.603i)9-s + (0.0747 + 0.997i)12-s + (−0.603 + 1.17i)13-s + (0.878 + 0.478i)16-s + (−0.222 − 0.974i)19-s + (0.365 + 0.930i)21-s + (0.921 + 0.388i)25-s + (0.826 + 0.563i)27-s + (0.980 + 0.198i)28-s + 1.39·31-s + (0.921 − 0.388i)36-s + (−1.60 + 1.09i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.999−0.00868i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00868i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.999−0.00868i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00868i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.999−0.00868i0.999 - 0.00868i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(1472,⋅)\chi_{2793} (1472, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.999−0.00868i)(2,\ 2793,\ (\ :0),\ 0.999 - 0.00868i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.57738510900.5773851090
L(12)L(\frac12) ≈\approx 0.57738510900.5773851090
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+(0.318+0.947i)T 1 + (0.318 + 0.947i)T
7 1+(0.998−0.0498i)T 1 + (0.998 - 0.0498i)T
19 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
good2 1+(0.969+0.246i)T2 1 + (0.969 + 0.246i)T^{2}
5 1+(−0.921−0.388i)T2 1 + (-0.921 - 0.388i)T^{2}
11 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
13 1+(0.603−1.17i)T+(−0.583−0.811i)T2 1 + (0.603 - 1.17i)T + (-0.583 - 0.811i)T^{2}
17 1+(−0.878+0.478i)T2 1 + (-0.878 + 0.478i)T^{2}
23 1+(0.853−0.521i)T2 1 + (0.853 - 0.521i)T^{2}
29 1+(0.318+0.947i)T2 1 + (0.318 + 0.947i)T^{2}
31 1−1.39T+T2 1 - 1.39T + T^{2}
37 1+(1.60−1.09i)T+(0.365−0.930i)T2 1 + (1.60 - 1.09i)T + (0.365 - 0.930i)T^{2}
41 1+(−0.921−0.388i)T2 1 + (-0.921 - 0.388i)T^{2}
43 1+(−1.36+0.831i)T+(0.456−0.889i)T2 1 + (-1.36 + 0.831i)T + (0.456 - 0.889i)T^{2}
47 1+(0.583+0.811i)T2 1 + (0.583 + 0.811i)T^{2}
53 1+(−0.878−0.478i)T2 1 + (-0.878 - 0.478i)T^{2}
59 1+(0.998−0.0498i)T2 1 + (0.998 - 0.0498i)T^{2}
61 1+(0.815−1.13i)T+(−0.318−0.947i)T2 1 + (0.815 - 1.13i)T + (-0.318 - 0.947i)T^{2}
67 1+(−1.24+0.452i)T+(0.766−0.642i)T2 1 + (-1.24 + 0.452i)T + (0.766 - 0.642i)T^{2}
71 1+(−0.980+0.198i)T2 1 + (-0.980 + 0.198i)T^{2}
73 1+(−0.293+0.455i)T+(−0.411−0.911i)T2 1 + (-0.293 + 0.455i)T + (-0.411 - 0.911i)T^{2}
79 1+(−0.345−1.95i)T+(−0.939+0.342i)T2 1 + (-0.345 - 1.95i)T + (-0.939 + 0.342i)T^{2}
83 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
89 1+(0.969−0.246i)T2 1 + (0.969 - 0.246i)T^{2}
97 1+(0.254+1.44i)T+(−0.939+0.342i)T2 1 + (0.254 + 1.44i)T + (-0.939 + 0.342i)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

−8.910727232323972317516818290221, −8.417494404990972301173773542721, −7.22694415243442312922331502486, −6.77983639450923847517581419097, −6.03483500302449925681683293373, −5.10900883469785557799520486524, −4.45169598391953047771064180093, −3.26615507090109794230970390369, −2.27239968447369710819652815882, −0.908200707154055944671163961136, 0.53755670340749483201835547106, 2.79947811119206093703313204460, 3.46010704498771148495438909861, 4.24437733571727832546668660550, 5.06347052563714553901965333053, 5.73069232977959836613621207126, 6.51626951367571897819637145987, 7.65808354675653694201231440357, 8.402267572006393296632377619629, 9.116017824367670115746089140467

Graph of the ZZ-function along the critical line