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Properties

Label 2-483-23.3-c1-0-18
Degree 22
Conductor 483483
Sign 0.999+0.0131i0.999 + 0.0131i
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.186 + 1.29i)2-s + (0.415 − 0.909i)3-s + (0.273 − 0.0801i)4-s + (1.65 − 1.90i)5-s + (1.25 + 0.368i)6-s + (0.841 − 0.540i)7-s + (1.24 + 2.72i)8-s + (−0.654 − 0.755i)9-s + (2.77 + 1.78i)10-s + (0.470 − 3.27i)11-s + (0.0405 − 0.281i)12-s + (−2.21 − 1.42i)13-s + (0.857 + 0.989i)14-s + (−1.04 − 2.29i)15-s + (−2.81 + 1.81i)16-s + (−2.87 − 0.843i)17-s + ⋯
L(s)  = 1  + (0.131 + 0.916i)2-s + (0.239 − 0.525i)3-s + (0.136 − 0.0400i)4-s + (0.738 − 0.852i)5-s + (0.513 + 0.150i)6-s + (0.317 − 0.204i)7-s + (0.439 + 0.962i)8-s + (−0.218 − 0.251i)9-s + (0.878 + 0.564i)10-s + (0.141 − 0.986i)11-s + (0.0116 − 0.0813i)12-s + (−0.615 − 0.395i)13-s + (0.229 + 0.264i)14-s + (−0.270 − 0.592i)15-s + (−0.704 + 0.452i)16-s + (−0.696 − 0.204i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.999+0.0131i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.999+0.0131i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.999+0.0131i0.999 + 0.0131i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(463,⋅)\chi_{483} (463, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.999+0.0131i)(2,\ 483,\ (\ :1/2),\ 0.999 + 0.0131i)

Particular Values

L(1)L(1) ≈\approx 2.07502−0.0136880i2.07502 - 0.0136880i
L(12)L(\frac12) ≈\approx 2.07502−0.0136880i2.07502 - 0.0136880i
L(32)L(\frac{3}{2}) 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+(−0.415+0.909i)T 1 + (-0.415 + 0.909i)T
7 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
23 1+(−4.63−1.24i)T 1 + (-4.63 - 1.24i)T
good2 1+(−0.186−1.29i)T+(−1.91+0.563i)T2 1 + (-0.186 - 1.29i)T + (-1.91 + 0.563i)T^{2}
5 1+(−1.65+1.90i)T+(−0.711−4.94i)T2 1 + (-1.65 + 1.90i)T + (-0.711 - 4.94i)T^{2}
11 1+(−0.470+3.27i)T+(−10.5−3.09i)T2 1 + (-0.470 + 3.27i)T + (-10.5 - 3.09i)T^{2}
13 1+(2.21+1.42i)T+(5.40+11.8i)T2 1 + (2.21 + 1.42i)T + (5.40 + 11.8i)T^{2}
17 1+(2.87+0.843i)T+(14.3+9.19i)T2 1 + (2.87 + 0.843i)T + (14.3 + 9.19i)T^{2}
19 1+(4.39−1.29i)T+(15.9−10.2i)T2 1 + (4.39 - 1.29i)T + (15.9 - 10.2i)T^{2}
29 1+(−8.79−2.58i)T+(24.3+15.6i)T2 1 + (-8.79 - 2.58i)T + (24.3 + 15.6i)T^{2}
31 1+(0.0842+0.184i)T+(−20.3+23.4i)T2 1 + (0.0842 + 0.184i)T + (-20.3 + 23.4i)T^{2}
37 1+(1.42+1.63i)T+(−5.26+36.6i)T2 1 + (1.42 + 1.63i)T + (-5.26 + 36.6i)T^{2}
41 1+(8.17−9.43i)T+(−5.83−40.5i)T2 1 + (8.17 - 9.43i)T + (-5.83 - 40.5i)T^{2}
43 1+(3.94−8.63i)T+(−28.1−32.4i)T2 1 + (3.94 - 8.63i)T + (-28.1 - 32.4i)T^{2}
47 1−5.57T+47T2 1 - 5.57T + 47T^{2}
53 1+(−8.67+5.57i)T+(22.0−48.2i)T2 1 + (-8.67 + 5.57i)T + (22.0 - 48.2i)T^{2}
59 1+(−1.24−0.800i)T+(24.5+53.6i)T2 1 + (-1.24 - 0.800i)T + (24.5 + 53.6i)T^{2}
61 1+(−1.38−3.03i)T+(−39.9+46.1i)T2 1 + (-1.38 - 3.03i)T + (-39.9 + 46.1i)T^{2}
67 1+(−1.58−11.0i)T+(−64.2+18.8i)T2 1 + (-1.58 - 11.0i)T + (-64.2 + 18.8i)T^{2}
71 1+(−1.08−7.53i)T+(−68.1+20.0i)T2 1 + (-1.08 - 7.53i)T + (-68.1 + 20.0i)T^{2}
73 1+(−4.67+1.37i)T+(61.4−39.4i)T2 1 + (-4.67 + 1.37i)T + (61.4 - 39.4i)T^{2}
79 1+(14.2+9.15i)T+(32.8+71.8i)T2 1 + (14.2 + 9.15i)T + (32.8 + 71.8i)T^{2}
83 1+(3.75+4.33i)T+(−11.8+82.1i)T2 1 + (3.75 + 4.33i)T + (-11.8 + 82.1i)T^{2}
89 1+(−5.48+12.0i)T+(−58.2−67.2i)T2 1 + (-5.48 + 12.0i)T + (-58.2 - 67.2i)T^{2}
97 1+(4.55−5.26i)T+(−13.8−96.0i)T2 1 + (4.55 - 5.26i)T + (-13.8 - 96.0i)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

−11.03342253051575294403904232342, −10.01214448436424253875122874567, −8.633815426172076502267935631024, −8.425260166276038398046383021823, −7.17008357996907649071297184231, −6.40567110014006875909810713390, −5.48833617109183890219692848179, −4.66976109054865085698520687607, −2.73332811052003537948259590766, −1.36727850234478730778449611855, 2.07913971279402450357802337394, 2.60282000933952006111123696448, 4.01347824809972523090618578441, 4.96228394547578409615514780591, 6.58561338655097336022482653670, 7.04959112165078693888268134538, 8.576322384558449647590287134900, 9.537838842952437638828251073265, 10.46939108887195713075508319550, 10.62575509024024573414754814327

Graph of the ZZ-function along the critical line