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Properties

Label 2-675-27.22-c1-0-20
Degree 22
Conductor 675675
Sign 0.532−0.846i0.532 - 0.846i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (1.02 − 0.860i)2-s + (0.817 + 1.52i)3-s + (−0.0359 + 0.204i)4-s + (2.15 + 0.862i)6-s + (0.0601 + 0.340i)7-s + (1.47 + 2.55i)8-s + (−1.66 + 2.49i)9-s + (−0.377 − 0.137i)11-s + (−0.341 + 0.111i)12-s + (−0.575 − 0.483i)13-s + (0.355 + 0.297i)14-s + (3.32 + 1.21i)16-s + (0.670 − 1.16i)17-s + (0.442 + 3.99i)18-s + (1.87 + 3.24i)19-s + ⋯
L(s)  = 1  + (0.725 − 0.608i)2-s + (0.471 + 0.881i)3-s + (−0.0179 + 0.102i)4-s + (0.878 + 0.352i)6-s + (0.0227 + 0.128i)7-s + (0.522 + 0.904i)8-s + (−0.554 + 0.832i)9-s + (−0.113 − 0.0414i)11-s + (−0.0984 + 0.0323i)12-s + (−0.159 − 0.133i)13-s + (0.0948 + 0.0796i)14-s + (0.832 + 0.302i)16-s + (0.162 − 0.281i)17-s + (0.104 + 0.940i)18-s + (0.429 + 0.744i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.532−0.846i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.532−0.846i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.532−0.846i0.532 - 0.846i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(76,⋅)\chi_{675} (76, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.532−0.846i)(2,\ 675,\ (\ :1/2),\ 0.532 - 0.846i)

Particular Values

L(1)L(1) ≈\approx 2.13173+1.17740i2.13173 + 1.17740i
L(12)L(\frac12) ≈\approx 2.13173+1.17740i2.13173 + 1.17740i
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+(−0.817−1.52i)T 1 + (-0.817 - 1.52i)T
5 1 1
good2 1+(−1.02+0.860i)T+(0.347−1.96i)T2 1 + (-1.02 + 0.860i)T + (0.347 - 1.96i)T^{2}
7 1+(−0.0601−0.340i)T+(−6.57+2.39i)T2 1 + (-0.0601 - 0.340i)T + (-6.57 + 2.39i)T^{2}
11 1+(0.377+0.137i)T+(8.42+7.07i)T2 1 + (0.377 + 0.137i)T + (8.42 + 7.07i)T^{2}
13 1+(0.575+0.483i)T+(2.25+12.8i)T2 1 + (0.575 + 0.483i)T + (2.25 + 12.8i)T^{2}
17 1+(−0.670+1.16i)T+(−8.5−14.7i)T2 1 + (-0.670 + 1.16i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.87−3.24i)T+(−9.5+16.4i)T2 1 + (-1.87 - 3.24i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.536−3.04i)T+(−21.6−7.86i)T2 1 + (0.536 - 3.04i)T + (-21.6 - 7.86i)T^{2}
29 1+(−3.79+3.18i)T+(5.03−28.5i)T2 1 + (-3.79 + 3.18i)T + (5.03 - 28.5i)T^{2}
31 1+(0.774−4.39i)T+(−29.1−10.6i)T2 1 + (0.774 - 4.39i)T + (-29.1 - 10.6i)T^{2}
37 1+(−1.25+2.16i)T+(−18.5−32.0i)T2 1 + (-1.25 + 2.16i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.73−1.45i)T+(7.11+40.3i)T2 1 + (-1.73 - 1.45i)T + (7.11 + 40.3i)T^{2}
43 1+(8.03+2.92i)T+(32.9+27.6i)T2 1 + (8.03 + 2.92i)T + (32.9 + 27.6i)T^{2}
47 1+(2.13+12.1i)T+(−44.1+16.0i)T2 1 + (2.13 + 12.1i)T + (-44.1 + 16.0i)T^{2}
53 1−10.1T+53T2 1 - 10.1T + 53T^{2}
59 1+(−12.7+4.64i)T+(45.1−37.9i)T2 1 + (-12.7 + 4.64i)T + (45.1 - 37.9i)T^{2}
61 1+(−2.31−13.1i)T+(−57.3+20.8i)T2 1 + (-2.31 - 13.1i)T + (-57.3 + 20.8i)T^{2}
67 1+(9.40+7.89i)T+(11.6+65.9i)T2 1 + (9.40 + 7.89i)T + (11.6 + 65.9i)T^{2}
71 1+(1.14−1.98i)T+(−35.5−61.4i)T2 1 + (1.14 - 1.98i)T + (-35.5 - 61.4i)T^{2}
73 1+(6.23+10.8i)T+(−36.5+63.2i)T2 1 + (6.23 + 10.8i)T + (-36.5 + 63.2i)T^{2}
79 1+(−9.11+7.64i)T+(13.7−77.7i)T2 1 + (-9.11 + 7.64i)T + (13.7 - 77.7i)T^{2}
83 1+(−3.71+3.11i)T+(14.4−81.7i)T2 1 + (-3.71 + 3.11i)T + (14.4 - 81.7i)T^{2}
89 1+(0.197+0.341i)T+(−44.5+77.0i)T2 1 + (0.197 + 0.341i)T + (-44.5 + 77.0i)T^{2}
97 1+(−13.9−5.07i)T+(74.3+62.3i)T2 1 + (-13.9 - 5.07i)T + (74.3 + 62.3i)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

−10.53898735963651999036162144577, −10.06339899104003695592990151405, −8.913684471919943670396581290230, −8.205085294431949057596639781252, −7.30352325256182788303050322806, −5.64743458532233604246053713054, −4.96359386071825799315445755358, −3.91853340170749070122896833616, −3.20270766564082093720628363547, −2.12599159912207851973369320870, 1.05491623349453128584434891938, 2.60199679454719406380133663904, 3.89763003808994751971578013375, 4.99142798520199354980952317442, 6.00838462334615200360175050166, 6.77522060470762273242906885647, 7.46615187096169703349550078335, 8.416240964278201662794907987073, 9.404947062562740564403581215558, 10.28624659272796074190820862780

Graph of the ZZ-function along the critical line