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Properties

Label 2-3e5-27.25-c1-0-4
Degree 22
Conductor 243243
Sign 0.978−0.207i0.978 - 0.207i
Analytic cond. 1.940361.94036
Root an. cond. 1.392961.39296
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  + (−0.0721 + 0.409i)2-s + (1.71 + 0.625i)4-s + (1.69 − 1.42i)5-s + (1.24 − 0.451i)7-s + (−0.795 + 1.37i)8-s + (0.460 + 0.797i)10-s + (−3.99 − 3.35i)11-s + (−0.00313 − 0.0177i)13-s + (0.0952 + 0.539i)14-s + (2.29 + 1.92i)16-s + (1.56 + 2.71i)17-s + (−0.208 + 0.361i)19-s + (3.80 − 1.38i)20-s + (1.65 − 1.39i)22-s + (0.972 + 0.353i)23-s + ⋯
L(s)  = 1  + (−0.0510 + 0.289i)2-s + (0.858 + 0.312i)4-s + (0.758 − 0.636i)5-s + (0.468 − 0.170i)7-s + (−0.281 + 0.486i)8-s + (0.145 + 0.252i)10-s + (−1.20 − 1.01i)11-s + (−0.000869 − 0.00493i)13-s + (0.0254 + 0.144i)14-s + (0.573 + 0.481i)16-s + (0.379 + 0.658i)17-s + (−0.0478 + 0.0829i)19-s + (0.850 − 0.309i)20-s + (0.353 − 0.296i)22-s + (0.202 + 0.0737i)23-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.978−0.207i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.978−0.207i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.978−0.207i0.978 - 0.207i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(217,⋅)\chi_{243} (217, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 0.978−0.207i)(2,\ 243,\ (\ :1/2),\ 0.978 - 0.207i)

Particular Values

L(1)L(1) ≈\approx 1.57375+0.164864i1.57375 + 0.164864i
L(12)L(\frac12) ≈\approx 1.57375+0.164864i1.57375 + 0.164864i
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 1
good2 1+(0.0721−0.409i)T+(−1.87−0.684i)T2 1 + (0.0721 - 0.409i)T + (-1.87 - 0.684i)T^{2}
5 1+(−1.69+1.42i)T+(0.868−4.92i)T2 1 + (-1.69 + 1.42i)T + (0.868 - 4.92i)T^{2}
7 1+(−1.24+0.451i)T+(5.36−4.49i)T2 1 + (-1.24 + 0.451i)T + (5.36 - 4.49i)T^{2}
11 1+(3.99+3.35i)T+(1.91+10.8i)T2 1 + (3.99 + 3.35i)T + (1.91 + 10.8i)T^{2}
13 1+(0.00313+0.0177i)T+(−12.2+4.44i)T2 1 + (0.00313 + 0.0177i)T + (-12.2 + 4.44i)T^{2}
17 1+(−1.56−2.71i)T+(−8.5+14.7i)T2 1 + (-1.56 - 2.71i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.208−0.361i)T+(−9.5−16.4i)T2 1 + (0.208 - 0.361i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.972−0.353i)T+(17.6+14.7i)T2 1 + (-0.972 - 0.353i)T + (17.6 + 14.7i)T^{2}
29 1+(1.35−7.68i)T+(−27.2−9.91i)T2 1 + (1.35 - 7.68i)T + (-27.2 - 9.91i)T^{2}
31 1+(3.50+1.27i)T+(23.7+19.9i)T2 1 + (3.50 + 1.27i)T + (23.7 + 19.9i)T^{2}
37 1+(2.21+3.83i)T+(−18.5+32.0i)T2 1 + (2.21 + 3.83i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.638+3.61i)T+(−38.5+14.0i)T2 1 + (0.638 + 3.61i)T + (-38.5 + 14.0i)T^{2}
43 1+(6.36+5.34i)T+(7.46+42.3i)T2 1 + (6.36 + 5.34i)T + (7.46 + 42.3i)T^{2}
47 1+(−6.66+2.42i)T+(36.0−30.2i)T2 1 + (-6.66 + 2.42i)T + (36.0 - 30.2i)T^{2}
53 1+1.30T+53T2 1 + 1.30T + 53T^{2}
59 1+(2.83−2.37i)T+(10.2−58.1i)T2 1 + (2.83 - 2.37i)T + (10.2 - 58.1i)T^{2}
61 1+(6.49−2.36i)T+(46.7−39.2i)T2 1 + (6.49 - 2.36i)T + (46.7 - 39.2i)T^{2}
67 1+(1.91+10.8i)T+(−62.9+22.9i)T2 1 + (1.91 + 10.8i)T + (-62.9 + 22.9i)T^{2}
71 1+(−3.04−5.26i)T+(−35.5+61.4i)T2 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.273+0.473i)T+(−36.5−63.2i)T2 1 + (-0.273 + 0.473i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.0849+0.481i)T+(−74.2−27.0i)T2 1 + (-0.0849 + 0.481i)T + (-74.2 - 27.0i)T^{2}
83 1+(−0.801+4.54i)T+(−77.9−28.3i)T2 1 + (-0.801 + 4.54i)T + (-77.9 - 28.3i)T^{2}
89 1+(−1.68+2.92i)T+(−44.5−77.0i)T2 1 + (-1.68 + 2.92i)T + (-44.5 - 77.0i)T^{2}
97 1+(−7.61−6.39i)T+(16.8+95.5i)T2 1 + (-7.61 - 6.39i)T + (16.8 + 95.5i)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

−12.24210792119864349831145307147, −10.98692033739861607334057548029, −10.50117215753426721455264522375, −9.011875593047442609782388299489, −8.175901958662447453885875449981, −7.25175054062979100950269671762, −5.87790044578573527147396171079, −5.24404054047464530077155546490, −3.31938531815546667277092779730, −1.80226328653260689950495554847, 1.96559335472103053662194687295, 2.88138239861605174587165751004, 4.92718968029792367314396753378, 6.00691629560190741522052037127, 7.05442492212553151321865045780, 7.948225830523985184087278566107, 9.639118961252779626702895114572, 10.17668127736507280942924757655, 11.03882312088696563738672828618, 11.91861731091374152413235578565

Graph of the ZZ-function along the critical line