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Properties

Label 2-405-135.122-c1-0-3
Degree 22
Conductor 405405
Sign 0.997+0.0729i0.997 + 0.0729i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (−1.23 − 0.866i)2-s + (0.0960 + 0.264i)4-s + (2.23 + 0.140i)5-s + (0.677 + 1.45i)7-s + (−0.671 + 2.50i)8-s + (−2.63 − 2.10i)10-s + (1.78 + 2.12i)11-s + (0.176 + 0.251i)13-s + (0.420 − 2.38i)14-s + (3.43 − 2.88i)16-s + (1.88 + 7.02i)17-s + (−5.07 + 2.92i)19-s + (0.177 + 0.602i)20-s + (−0.365 − 4.17i)22-s + (−0.116 − 0.0543i)23-s + ⋯
L(s)  = 1  + (−0.874 − 0.612i)2-s + (0.0480 + 0.132i)4-s + (0.998 + 0.0626i)5-s + (0.256 + 0.549i)7-s + (−0.237 + 0.886i)8-s + (−0.834 − 0.666i)10-s + (0.538 + 0.641i)11-s + (0.0489 + 0.0698i)13-s + (0.112 − 0.637i)14-s + (0.858 − 0.720i)16-s + (0.456 + 1.70i)17-s + (−1.16 + 0.671i)19-s + (0.0396 + 0.134i)20-s + (−0.0779 − 0.890i)22-s + (−0.0243 − 0.0113i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((0.997+0.0729i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0729i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((0.997+0.0729i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 0.997+0.0729i0.997 + 0.0729i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(152,⋅)\chi_{405} (152, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), 0.997+0.0729i)(2,\ 405,\ (\ :1/2),\ 0.997 + 0.0729i)

Particular Values

L(1)L(1) ≈\approx 1.01184−0.0369660i1.01184 - 0.0369660i
L(12)L(\frac12) ≈\approx 1.01184−0.0369660i1.01184 - 0.0369660i
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
5 1+(−2.23−0.140i)T 1 + (-2.23 - 0.140i)T
good2 1+(1.23+0.866i)T+(0.684+1.87i)T2 1 + (1.23 + 0.866i)T + (0.684 + 1.87i)T^{2}
7 1+(−0.677−1.45i)T+(−4.49+5.36i)T2 1 + (-0.677 - 1.45i)T + (-4.49 + 5.36i)T^{2}
11 1+(−1.78−2.12i)T+(−1.91+10.8i)T2 1 + (-1.78 - 2.12i)T + (-1.91 + 10.8i)T^{2}
13 1+(−0.176−0.251i)T+(−4.44+12.2i)T2 1 + (-0.176 - 0.251i)T + (-4.44 + 12.2i)T^{2}
17 1+(−1.88−7.02i)T+(−14.7+8.5i)T2 1 + (-1.88 - 7.02i)T + (-14.7 + 8.5i)T^{2}
19 1+(5.07−2.92i)T+(9.5−16.4i)T2 1 + (5.07 - 2.92i)T + (9.5 - 16.4i)T^{2}
23 1+(0.116+0.0543i)T+(14.7+17.6i)T2 1 + (0.116 + 0.0543i)T + (14.7 + 17.6i)T^{2}
29 1+(1.51+8.60i)T+(−27.2+9.91i)T2 1 + (1.51 + 8.60i)T + (-27.2 + 9.91i)T^{2}
31 1+(−4.22+1.53i)T+(23.7−19.9i)T2 1 + (-4.22 + 1.53i)T + (23.7 - 19.9i)T^{2}
37 1+(−2.08+0.558i)T+(32.0−18.5i)T2 1 + (-2.08 + 0.558i)T + (32.0 - 18.5i)T^{2}
41 1+(−4.62−0.815i)T+(38.5+14.0i)T2 1 + (-4.62 - 0.815i)T + (38.5 + 14.0i)T^{2}
43 1+(0.429−4.90i)T+(−42.3−7.46i)T2 1 + (0.429 - 4.90i)T + (-42.3 - 7.46i)T^{2}
47 1+(−7.69+3.58i)T+(30.2−36.0i)T2 1 + (-7.69 + 3.58i)T + (30.2 - 36.0i)T^{2}
53 1+(0.483+0.483i)T+53iT2 1 + (0.483 + 0.483i)T + 53iT^{2}
59 1+(7.43+6.23i)T+(10.2+58.1i)T2 1 + (7.43 + 6.23i)T + (10.2 + 58.1i)T^{2}
61 1+(−2.50−0.910i)T+(46.7+39.2i)T2 1 + (-2.50 - 0.910i)T + (46.7 + 39.2i)T^{2}
67 1+(0.884−0.619i)T+(22.9−62.9i)T2 1 + (0.884 - 0.619i)T + (22.9 - 62.9i)T^{2}
71 1+(9.03+5.21i)T+(35.5+61.4i)T2 1 + (9.03 + 5.21i)T + (35.5 + 61.4i)T^{2}
73 1+(6.65+1.78i)T+(63.2+36.5i)T2 1 + (6.65 + 1.78i)T + (63.2 + 36.5i)T^{2}
79 1+(−2.04+0.360i)T+(74.2−27.0i)T2 1 + (-2.04 + 0.360i)T + (74.2 - 27.0i)T^{2}
83 1+(7.55−10.7i)T+(−28.3−77.9i)T2 1 + (7.55 - 10.7i)T + (-28.3 - 77.9i)T^{2}
89 1+(−5.58−9.66i)T+(−44.5+77.0i)T2 1 + (-5.58 - 9.66i)T + (-44.5 + 77.0i)T^{2}
97 1+(11.8+1.03i)T+(95.5+16.8i)T2 1 + (11.8 + 1.03i)T + (95.5 + 16.8i)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

−10.94978953432935195365099606306, −10.19551161472311390621711436043, −9.605861325904057947697840866392, −8.707624913431797494902189933301, −7.963998164760337788392798433220, −6.28654883550746059867467762382, −5.72312925631643775027008076996, −4.25868053698845498193951442147, −2.38002499542498979933942664669, −1.57589829850090026820791324647, 0.997468405866986239595126058438, 2.94399926529676460878386637414, 4.47930976027817639198192466237, 5.79205702960526189549181450036, 6.79844321737405234907189323156, 7.47600435812684001249750872067, 8.795933744966146792501165982043, 9.120571263233791659293045776500, 10.17135610283004046741001158902, 10.94058328076309911998894802906

Graph of the ZZ-function along the critical line