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Properties

Label 2-189-189.104-c1-0-12
Degree 22
Conductor 189189
Sign 0.994−0.101i0.994 - 0.101i
Analytic cond. 1.509171.50917
Root an. cond. 1.228481.22848
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  + (−0.0448 + 0.123i)2-s + (1.62 − 0.587i)3-s + (1.51 + 1.27i)4-s + (0.231 − 1.31i)5-s + (−0.000699 + 0.227i)6-s + (−0.772 + 2.53i)7-s + (−0.452 + 0.261i)8-s + (2.30 − 1.91i)9-s + (0.151 + 0.0873i)10-s + (−4.31 + 0.759i)11-s + (3.22 + 1.18i)12-s + (−1.38 − 3.81i)13-s + (−0.277 − 0.208i)14-s + (−0.393 − 2.27i)15-s + (0.676 + 3.83i)16-s + (1.85 − 3.21i)17-s + ⋯
L(s)  = 1  + (−0.0317 + 0.0871i)2-s + (0.940 − 0.339i)3-s + (0.759 + 0.637i)4-s + (0.103 − 0.586i)5-s + (−0.000285 + 0.0927i)6-s + (−0.292 + 0.956i)7-s + (−0.159 + 0.0923i)8-s + (0.769 − 0.638i)9-s + (0.0478 + 0.0276i)10-s + (−1.29 + 0.229i)11-s + (0.930 + 0.341i)12-s + (−0.385 − 1.05i)13-s + (−0.0740 − 0.0557i)14-s + (−0.101 − 0.587i)15-s + (0.169 + 0.959i)16-s + (0.450 − 0.780i)17-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((0.994−0.101i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((0.994−0.101i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: 0.994−0.101i0.994 - 0.101i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(104,⋅)\chi_{189} (104, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), 0.994−0.101i)(2,\ 189,\ (\ :1/2),\ 0.994 - 0.101i)

Particular Values

L(1)L(1) ≈\approx 1.64599+0.0837184i1.64599 + 0.0837184i
L(12)L(\frac12) ≈\approx 1.64599+0.0837184i1.64599 + 0.0837184i
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+(−1.62+0.587i)T 1 + (-1.62 + 0.587i)T
7 1+(0.772−2.53i)T 1 + (0.772 - 2.53i)T
good2 1+(0.0448−0.123i)T+(−1.53−1.28i)T2 1 + (0.0448 - 0.123i)T + (-1.53 - 1.28i)T^{2}
5 1+(−0.231+1.31i)T+(−4.69−1.71i)T2 1 + (-0.231 + 1.31i)T + (-4.69 - 1.71i)T^{2}
11 1+(4.31−0.759i)T+(10.3−3.76i)T2 1 + (4.31 - 0.759i)T + (10.3 - 3.76i)T^{2}
13 1+(1.38+3.81i)T+(−9.95+8.35i)T2 1 + (1.38 + 3.81i)T + (-9.95 + 8.35i)T^{2}
17 1+(−1.85+3.21i)T+(−8.5−14.7i)T2 1 + (-1.85 + 3.21i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.31−2.49i)T+(9.5−16.4i)T2 1 + (4.31 - 2.49i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.242+0.289i)T+(−3.99−22.6i)T2 1 + (-0.242 + 0.289i)T + (-3.99 - 22.6i)T^{2}
29 1+(−1.57+4.31i)T+(−22.2−18.6i)T2 1 + (-1.57 + 4.31i)T + (-22.2 - 18.6i)T^{2}
31 1+(0.693−0.826i)T+(−5.38−30.5i)T2 1 + (0.693 - 0.826i)T + (-5.38 - 30.5i)T^{2}
37 1+(0.172−0.298i)T+(−18.5−32.0i)T2 1 + (0.172 - 0.298i)T + (-18.5 - 32.0i)T^{2}
41 1+(5.09−1.85i)T+(31.4−26.3i)T2 1 + (5.09 - 1.85i)T + (31.4 - 26.3i)T^{2}
43 1+(0.390+2.21i)T+(−40.4+14.7i)T2 1 + (0.390 + 2.21i)T + (-40.4 + 14.7i)T^{2}
47 1+(9.77−8.19i)T+(8.16−46.2i)T2 1 + (9.77 - 8.19i)T + (8.16 - 46.2i)T^{2}
53 1−9.19iT−53T2 1 - 9.19iT - 53T^{2}
59 1+(−2.24+12.7i)T+(−55.4−20.1i)T2 1 + (-2.24 + 12.7i)T + (-55.4 - 20.1i)T^{2}
61 1+(−1.14−1.36i)T+(−10.5+60.0i)T2 1 + (-1.14 - 1.36i)T + (-10.5 + 60.0i)T^{2}
67 1+(−5.40+1.96i)T+(51.3−43.0i)T2 1 + (-5.40 + 1.96i)T + (51.3 - 43.0i)T^{2}
71 1+(−2.30−1.33i)T+(35.5+61.4i)T2 1 + (-2.30 - 1.33i)T + (35.5 + 61.4i)T^{2}
73 1+(−4.63+2.67i)T+(36.5−63.2i)T2 1 + (-4.63 + 2.67i)T + (36.5 - 63.2i)T^{2}
79 1+(−5.48−1.99i)T+(60.5+50.7i)T2 1 + (-5.48 - 1.99i)T + (60.5 + 50.7i)T^{2}
83 1+(−4.03−1.46i)T+(63.5+53.3i)T2 1 + (-4.03 - 1.46i)T + (63.5 + 53.3i)T^{2}
89 1+(−5.59−9.68i)T+(−44.5+77.0i)T2 1 + (-5.59 - 9.68i)T + (-44.5 + 77.0i)T^{2}
97 1+(−14.5+2.55i)T+(91.1−33.1i)T2 1 + (-14.5 + 2.55i)T + (91.1 - 33.1i)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

−12.72782176639392746729505753878, −12.00931297162052718351910105323, −10.50762017424586084353974482866, −9.422078589720078081359003515473, −8.228352880009328846003913396475, −7.84267183576469791706348658329, −6.48975508753989288984267554583, −5.11927155393591232900286323589, −3.17674697030904953857823489889, −2.30222797095000372415086318635, 2.12215285300381282511656673305, 3.34862435027317260679713836122, 4.88292153820271451847752433299, 6.58415489858408926407252366312, 7.29924559421858115762824651727, 8.525593773459567715502415520573, 9.952836705718468247638983160329, 10.40647498631566529849348186544, 11.17203306311286477485808565376, 12.76110322068616249917619672982

Graph of the ZZ-function along the critical line