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Properties

Label 2-2793-2793.185-c0-0-0
Degree 22
Conductor 27932793
Sign 0.233−0.972i0.233 - 0.972i
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  + (0.0249 − 0.999i)3-s + (0.583 + 0.811i)4-s + (−0.124 + 0.992i)7-s + (−0.998 − 0.0498i)9-s + (0.826 − 0.563i)12-s + (−1.57 + 0.662i)13-s + (−0.318 + 0.947i)16-s + (0.222 + 0.974i)19-s + (0.988 + 0.149i)21-s + (−0.542 + 0.840i)25-s + (−0.0747 + 0.997i)27-s + (−0.878 + 0.478i)28-s − 0.822·31-s + (−0.542 − 0.840i)36-s + (1.61 − 0.121i)37-s + ⋯
L(s)  = 1  + (0.0249 − 0.999i)3-s + (0.583 + 0.811i)4-s + (−0.124 + 0.992i)7-s + (−0.998 − 0.0498i)9-s + (0.826 − 0.563i)12-s + (−1.57 + 0.662i)13-s + (−0.318 + 0.947i)16-s + (0.222 + 0.974i)19-s + (0.988 + 0.149i)21-s + (−0.542 + 0.840i)25-s + (−0.0747 + 0.997i)27-s + (−0.878 + 0.478i)28-s − 0.822·31-s + (−0.542 − 0.840i)36-s + (1.61 − 0.121i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.233−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.233−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.233−0.972i0.233 - 0.972i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(185,⋅)\chi_{2793} (185, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.233−0.972i)(2,\ 2793,\ (\ :0),\ 0.233 - 0.972i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0344675421.034467542
L(12)L(\frac12) ≈\approx 1.0344675421.034467542
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+(−0.0249+0.999i)T 1 + (-0.0249 + 0.999i)T
7 1+(0.124−0.992i)T 1 + (0.124 - 0.992i)T
19 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
good2 1+(−0.583−0.811i)T2 1 + (-0.583 - 0.811i)T^{2}
5 1+(0.542−0.840i)T2 1 + (0.542 - 0.840i)T^{2}
11 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
13 1+(1.57−0.662i)T+(0.698−0.715i)T2 1 + (1.57 - 0.662i)T + (0.698 - 0.715i)T^{2}
17 1+(−0.318−0.947i)T2 1 + (-0.318 - 0.947i)T^{2}
23 1+(−0.980+0.198i)T2 1 + (-0.980 + 0.198i)T^{2}
29 1+(−0.0249+0.999i)T2 1 + (-0.0249 + 0.999i)T^{2}
31 1+0.822T+T2 1 + 0.822T + T^{2}
37 1+(−1.61+0.121i)T+(0.988−0.149i)T2 1 + (-1.61 + 0.121i)T + (0.988 - 0.149i)T^{2}
41 1+(−0.542+0.840i)T2 1 + (-0.542 + 0.840i)T^{2}
43 1+(−1.95+0.395i)T+(0.921−0.388i)T2 1 + (-1.95 + 0.395i)T + (0.921 - 0.388i)T^{2}
47 1+(0.698−0.715i)T2 1 + (0.698 - 0.715i)T^{2}
53 1+(−0.318+0.947i)T2 1 + (-0.318 + 0.947i)T^{2}
59 1+(0.124−0.992i)T2 1 + (0.124 - 0.992i)T^{2}
61 1+(1.30−1.27i)T+(0.0249−0.999i)T2 1 + (1.30 - 1.27i)T + (0.0249 - 0.999i)T^{2}
67 1+(0.670+0.798i)T+(−0.173+0.984i)T2 1 + (0.670 + 0.798i)T + (-0.173 + 0.984i)T^{2}
71 1+(0.878+0.478i)T2 1 + (0.878 + 0.478i)T^{2}
73 1+(−0.120−0.158i)T+(−0.270+0.962i)T2 1 + (-0.120 - 0.158i)T + (-0.270 + 0.962i)T^{2}
79 1+(−0.168−0.463i)T+(−0.766+0.642i)T2 1 + (-0.168 - 0.463i)T + (-0.766 + 0.642i)T^{2}
83 1+(0.826−0.563i)T2 1 + (0.826 - 0.563i)T^{2}
89 1+(0.583−0.811i)T2 1 + (0.583 - 0.811i)T^{2}
97 1+(−1.79+0.653i)T+(0.766−0.642i)T2 1 + (-1.79 + 0.653i)T + (0.766 - 0.642i)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

−9.098063266831098934096939044882, −8.176682460683378091876123818378, −7.49630702712270607314084254984, −7.15693274626707159299792725478, −6.09138333944593411420246672798, −5.63258215108983014161306039455, −4.38584428458752071621665934394, −3.21184039246656447162758041864, −2.43656229732948460514272421017, −1.80138638100576939772116150391, 0.59931913364484919519524697518, 2.33479565731547478357625014926, 3.05602476662605820196293592981, 4.30018392698516630773058599200, 4.83124006063160143150845831932, 5.67246986156023508697751626543, 6.44301048326637958347738234436, 7.41789616757964152372429080630, 7.86135216381556640079250495341, 9.287650258841981754226888512181

Graph of the ZZ-function along the critical line