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Properties

Label 2-189-189.16-c1-0-0
Degree 22
Conductor 189189
Sign −0.0900+0.995i-0.0900 + 0.995i
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  + (−1.65 + 0.603i)2-s + (−1.50 + 0.853i)3-s + (0.851 − 0.714i)4-s + (−3.19 + 2.67i)5-s + (1.98 − 2.32i)6-s + (0.729 + 2.54i)7-s + (0.783 − 1.35i)8-s + (1.54 − 2.57i)9-s + (3.67 − 6.36i)10-s + (−1.89 − 1.58i)11-s + (−0.673 + 1.80i)12-s + (0.108 − 0.0907i)13-s + (−2.74 − 3.77i)14-s + (2.52 − 6.75i)15-s + (−0.866 + 4.91i)16-s + (−0.351 + 0.608i)17-s + ⋯
L(s)  = 1  + (−1.17 + 0.426i)2-s + (−0.870 + 0.492i)3-s + (0.425 − 0.357i)4-s + (−1.42 + 1.19i)5-s + (0.809 − 0.948i)6-s + (0.275 + 0.961i)7-s + (0.277 − 0.479i)8-s + (0.514 − 0.857i)9-s + (1.16 − 2.01i)10-s + (−0.570 − 0.478i)11-s + (−0.194 + 0.520i)12-s + (0.0299 − 0.0251i)13-s + (−0.733 − 1.00i)14-s + (0.651 − 1.74i)15-s + (−0.216 + 1.22i)16-s + (−0.0852 + 0.147i)17-s + ⋯

Functional equation

Λ(s)=(189s/2 ΓC(s) L(s)=((−0.0900+0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0900 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(189s/2 ΓC(s+1/2) L(s)=((−0.0900+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0900 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 189189    =    33⋅73^{3} \cdot 7
Sign: −0.0900+0.995i-0.0900 + 0.995i
Analytic conductor: 1.509171.50917
Root analytic conductor: 1.228481.22848
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ189(16,⋅)\chi_{189} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 189, ( :1/2), −0.0900+0.995i)(2,\ 189,\ (\ :1/2),\ -0.0900 + 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.0314116−0.0343786i0.0314116 - 0.0343786i
L(12)L(\frac12) ≈\approx 0.0314116−0.0343786i0.0314116 - 0.0343786i
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+(1.50−0.853i)T 1 + (1.50 - 0.853i)T
7 1+(−0.729−2.54i)T 1 + (-0.729 - 2.54i)T
good2 1+(1.65−0.603i)T+(1.53−1.28i)T2 1 + (1.65 - 0.603i)T + (1.53 - 1.28i)T^{2}
5 1+(3.19−2.67i)T+(0.868−4.92i)T2 1 + (3.19 - 2.67i)T + (0.868 - 4.92i)T^{2}
11 1+(1.89+1.58i)T+(1.91+10.8i)T2 1 + (1.89 + 1.58i)T + (1.91 + 10.8i)T^{2}
13 1+(−0.108+0.0907i)T+(2.25−12.8i)T2 1 + (-0.108 + 0.0907i)T + (2.25 - 12.8i)T^{2}
17 1+(0.351−0.608i)T+(−8.5−14.7i)T2 1 + (0.351 - 0.608i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.23+5.59i)T+(−9.5+16.4i)T2 1 + (3.23 + 5.59i)T + (-9.5 + 16.4i)T^{2}
23 1+(−5.24−1.90i)T+(17.6+14.7i)T2 1 + (-5.24 - 1.90i)T + (17.6 + 14.7i)T^{2}
29 1+(−0.100−0.0846i)T+(5.03+28.5i)T2 1 + (-0.100 - 0.0846i)T + (5.03 + 28.5i)T^{2}
31 1+(3.55−2.98i)T+(5.38−30.5i)T2 1 + (3.55 - 2.98i)T + (5.38 - 30.5i)T^{2}
37 1−0.775T+37T2 1 - 0.775T + 37T^{2}
41 1+(2.52−2.11i)T+(7.11−40.3i)T2 1 + (2.52 - 2.11i)T + (7.11 - 40.3i)T^{2}
43 1+(5.66−2.06i)T+(32.9−27.6i)T2 1 + (5.66 - 2.06i)T + (32.9 - 27.6i)T^{2}
47 1+(4.55+3.82i)T+(8.16+46.2i)T2 1 + (4.55 + 3.82i)T + (8.16 + 46.2i)T^{2}
53 1+(−1.85−3.22i)T+(−26.5+45.8i)T2 1 + (-1.85 - 3.22i)T + (-26.5 + 45.8i)T^{2}
59 1+(0.630+3.57i)T+(−55.4+20.1i)T2 1 + (0.630 + 3.57i)T + (-55.4 + 20.1i)T^{2}
61 1+(0.167+0.140i)T+(10.5+60.0i)T2 1 + (0.167 + 0.140i)T + (10.5 + 60.0i)T^{2}
67 1+(14.4+5.24i)T+(51.3+43.0i)T2 1 + (14.4 + 5.24i)T + (51.3 + 43.0i)T^{2}
71 1+(7.04+12.1i)T+(−35.5+61.4i)T2 1 + (7.04 + 12.1i)T + (-35.5 + 61.4i)T^{2}
73 1−12.7T+73T2 1 - 12.7T + 73T^{2}
79 1+(5.42−1.97i)T+(60.5−50.7i)T2 1 + (5.42 - 1.97i)T + (60.5 - 50.7i)T^{2}
83 1+(1.84+1.54i)T+(14.4+81.7i)T2 1 + (1.84 + 1.54i)T + (14.4 + 81.7i)T^{2}
89 1+(0.452+0.783i)T+(−44.5+77.0i)T2 1 + (0.452 + 0.783i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.73−0.629i)T+(74.3−62.3i)T2 1 + (1.73 - 0.629i)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

−13.02452790301848327741231206614, −11.83289297834584266443248940012, −11.03622417442030176596398807112, −10.55665559849358080990760615234, −9.181861056452167107550727429582, −8.281072932213606609000525573980, −7.22012743355082839349727996137, −6.40255222980157362701567871753, −4.76860151651763011349991498442, −3.27497301617792225114627679340, 0.07345063377660604240552211523, 1.39657334245255938382426908215, 4.26232178624134921107326202822, 5.16440054555445086763370993364, 7.19289206033700267199008827342, 7.87702009978334171145136951731, 8.623686721674459404310725614012, 10.05103431775139879391125637991, 10.90761088950754078561799483576, 11.62365268488874694913164144563

Graph of the ZZ-function along the critical line