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Properties

Label 2-405-135.113-c1-0-8
Degree 22
Conductor 405405
Sign 0.219+0.975i0.219 + 0.975i
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  + (−0.160 − 0.0140i)2-s + (−1.94 − 0.342i)4-s + (0.476 + 2.18i)5-s + (−2.37 − 1.66i)7-s + (0.617 + 0.165i)8-s + (−0.0456 − 0.356i)10-s + (1.48 − 4.09i)11-s + (−0.412 − 4.71i)13-s + (0.356 + 0.299i)14-s + (3.61 + 1.31i)16-s + (5.66 − 1.51i)17-s + (0.695 − 0.401i)19-s + (−0.176 − 4.41i)20-s + (−0.296 + 0.635i)22-s + (−1.52 − 2.18i)23-s + ⋯
L(s)  = 1  + (−0.113 − 0.00991i)2-s + (−0.972 − 0.171i)4-s + (0.212 + 0.977i)5-s + (−0.896 − 0.627i)7-s + (0.218 + 0.0585i)8-s + (−0.0144 − 0.112i)10-s + (0.449 − 1.23i)11-s + (−0.114 − 1.30i)13-s + (0.0953 + 0.0800i)14-s + (0.903 + 0.328i)16-s + (1.37 − 0.368i)17-s + (0.159 − 0.0920i)19-s + (−0.0394 − 0.986i)20-s + (−0.0631 + 0.135i)22-s + (−0.318 − 0.455i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((0.219+0.975i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((0.219+0.975i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 0.219+0.975i0.219 + 0.975i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(368,⋅)\chi_{405} (368, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), 0.219+0.975i)(2,\ 405,\ (\ :1/2),\ 0.219 + 0.975i)

Particular Values

L(1)L(1) ≈\approx 0.646582−0.517042i0.646582 - 0.517042i
L(12)L(\frac12) ≈\approx 0.646582−0.517042i0.646582 - 0.517042i
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+(−0.476−2.18i)T 1 + (-0.476 - 2.18i)T
good2 1+(0.160+0.0140i)T+(1.96+0.347i)T2 1 + (0.160 + 0.0140i)T + (1.96 + 0.347i)T^{2}
7 1+(2.37+1.66i)T+(2.39+6.57i)T2 1 + (2.37 + 1.66i)T + (2.39 + 6.57i)T^{2}
11 1+(−1.48+4.09i)T+(−8.42−7.07i)T2 1 + (-1.48 + 4.09i)T + (-8.42 - 7.07i)T^{2}
13 1+(0.412+4.71i)T+(−12.8+2.25i)T2 1 + (0.412 + 4.71i)T + (-12.8 + 2.25i)T^{2}
17 1+(−5.66+1.51i)T+(14.7−8.5i)T2 1 + (-5.66 + 1.51i)T + (14.7 - 8.5i)T^{2}
19 1+(−0.695+0.401i)T+(9.5−16.4i)T2 1 + (-0.695 + 0.401i)T + (9.5 - 16.4i)T^{2}
23 1+(1.52+2.18i)T+(−7.86+21.6i)T2 1 + (1.52 + 2.18i)T + (-7.86 + 21.6i)T^{2}
29 1+(2.06−1.73i)T+(5.03−28.5i)T2 1 + (2.06 - 1.73i)T + (5.03 - 28.5i)T^{2}
31 1+(−1.27+7.23i)T+(−29.1−10.6i)T2 1 + (-1.27 + 7.23i)T + (-29.1 - 10.6i)T^{2}
37 1+(0.841+3.14i)T+(−32.0+18.5i)T2 1 + (0.841 + 3.14i)T + (-32.0 + 18.5i)T^{2}
41 1+(2.88−3.44i)T+(−7.11−40.3i)T2 1 + (2.88 - 3.44i)T + (-7.11 - 40.3i)T^{2}
43 1+(2.90+6.22i)T+(−27.6+32.9i)T2 1 + (2.90 + 6.22i)T + (-27.6 + 32.9i)T^{2}
47 1+(3.17−4.52i)T+(−16.0−44.1i)T2 1 + (3.17 - 4.52i)T + (-16.0 - 44.1i)T^{2}
53 1+(−1.70+1.70i)T−53iT2 1 + (-1.70 + 1.70i)T - 53iT^{2}
59 1+(−4.31+1.57i)T+(45.1−37.9i)T2 1 + (-4.31 + 1.57i)T + (45.1 - 37.9i)T^{2}
61 1+(−2.10−11.9i)T+(−57.3+20.8i)T2 1 + (-2.10 - 11.9i)T + (-57.3 + 20.8i)T^{2}
67 1+(6.65−0.582i)T+(65.9−11.6i)T2 1 + (6.65 - 0.582i)T + (65.9 - 11.6i)T^{2}
71 1+(−5.61−3.24i)T+(35.5+61.4i)T2 1 + (-5.61 - 3.24i)T + (35.5 + 61.4i)T^{2}
73 1+(−2.49+9.31i)T+(−63.2−36.5i)T2 1 + (-2.49 + 9.31i)T + (-63.2 - 36.5i)T^{2}
79 1+(0.322+0.384i)T+(−13.7+77.7i)T2 1 + (0.322 + 0.384i)T + (-13.7 + 77.7i)T^{2}
83 1+(−0.780+8.92i)T+(−81.7−14.4i)T2 1 + (-0.780 + 8.92i)T + (-81.7 - 14.4i)T^{2}
89 1+(−6.84−11.8i)T+(−44.5+77.0i)T2 1 + (-6.84 - 11.8i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.60+3.08i)T+(62.3−74.3i)T2 1 + (-6.60 + 3.08i)T + (62.3 - 74.3i)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

−10.73630968574037110066622498677, −10.12423858420807865382457283738, −9.490291589240479603252970069403, −8.274718093115253550453149775014, −7.41046112467796865700899710586, −6.17010048084521429011255991933, −5.45603086279410905249743314479, −3.75115298587931973406005895761, −3.12141847978682974822897515253, −0.61305738115205673928502334144, 1.61813760210631701019077345371, 3.58288404823801596334033927082, 4.62323235853229240098812607293, 5.51729514472619127944987085046, 6.73017359213762821272766035573, 7.979414662881680603623343267765, 8.920139644466763970508939451887, 9.625542388432066927957824214399, 9.961123197701585678833010629423, 11.92907520047227523909770461918

Graph of the ZZ-function along the critical line