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Properties

Label 2-162-81.16-c1-0-4
Degree 22
Conductor 162162
Sign 0.866+0.499i0.866 + 0.499i
Analytic cond. 1.293571.29357
Root an. cond. 1.137351.13735
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  + (−0.973 − 0.230i)2-s + (1.56 − 0.739i)3-s + (0.893 + 0.448i)4-s + (0.149 − 0.201i)5-s + (−1.69 + 0.358i)6-s + (1.83 + 1.20i)7-s + (−0.766 − 0.642i)8-s + (1.90 − 2.31i)9-s + (−0.192 + 0.161i)10-s + (−1.53 + 0.179i)11-s + (1.73 + 0.0416i)12-s + (−0.160 + 0.536i)13-s + (−1.51 − 1.60i)14-s + (0.0858 − 0.426i)15-s + (0.597 + 0.802i)16-s + (−0.434 − 2.46i)17-s + ⋯
L(s)  = 1  + (−0.688 − 0.163i)2-s + (0.904 − 0.427i)3-s + (0.446 + 0.224i)4-s + (0.0670 − 0.0900i)5-s + (−0.691 + 0.146i)6-s + (0.695 + 0.457i)7-s + (−0.270 − 0.227i)8-s + (0.635 − 0.772i)9-s + (−0.0608 + 0.0510i)10-s + (−0.463 + 0.0541i)11-s + (0.499 + 0.0120i)12-s + (−0.0445 + 0.148i)13-s + (−0.403 − 0.428i)14-s + (0.0221 − 0.110i)15-s + (0.149 + 0.200i)16-s + (−0.105 − 0.597i)17-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.866+0.499i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+1/2) L(s)=((0.866+0.499i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.866+0.499i0.866 + 0.499i
Analytic conductor: 1.293571.29357
Root analytic conductor: 1.137351.13735
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ162(97,⋅)\chi_{162} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :1/2), 0.866+0.499i)(2,\ 162,\ (\ :1/2),\ 0.866 + 0.499i)

Particular Values

L(1)L(1) ≈\approx 1.11495−0.298357i1.11495 - 0.298357i
L(12)L(\frac12) ≈\approx 1.11495−0.298357i1.11495 - 0.298357i
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)
bad2 1+(0.973+0.230i)T 1 + (0.973 + 0.230i)T
3 1+(−1.56+0.739i)T 1 + (-1.56 + 0.739i)T
good5 1+(−0.149+0.201i)T+(−1.43−4.78i)T2 1 + (-0.149 + 0.201i)T + (-1.43 - 4.78i)T^{2}
7 1+(−1.83−1.20i)T+(2.77+6.42i)T2 1 + (-1.83 - 1.20i)T + (2.77 + 6.42i)T^{2}
11 1+(1.53−0.179i)T+(10.7−2.53i)T2 1 + (1.53 - 0.179i)T + (10.7 - 2.53i)T^{2}
13 1+(0.160−0.536i)T+(−10.8−7.14i)T2 1 + (0.160 - 0.536i)T + (-10.8 - 7.14i)T^{2}
17 1+(0.434+2.46i)T+(−15.9+5.81i)T2 1 + (0.434 + 2.46i)T + (-15.9 + 5.81i)T^{2}
19 1+(−0.461+2.61i)T+(−17.8−6.49i)T2 1 + (-0.461 + 2.61i)T + (-17.8 - 6.49i)T^{2}
23 1+(−0.530+0.349i)T+(9.10−21.1i)T2 1 + (-0.530 + 0.349i)T + (9.10 - 21.1i)T^{2}
29 1+(4.73−5.01i)T+(−1.68−28.9i)T2 1 + (4.73 - 5.01i)T + (-1.68 - 28.9i)T^{2}
31 1+(−0.0928−1.59i)T+(−30.7+3.59i)T2 1 + (-0.0928 - 1.59i)T + (-30.7 + 3.59i)T^{2}
37 1+(9.81+3.57i)T+(28.3+23.7i)T2 1 + (9.81 + 3.57i)T + (28.3 + 23.7i)T^{2}
41 1+(7.44−1.76i)T+(36.6−18.4i)T2 1 + (7.44 - 1.76i)T + (36.6 - 18.4i)T^{2}
43 1+(3.30−7.65i)T+(−29.5−31.2i)T2 1 + (3.30 - 7.65i)T + (-29.5 - 31.2i)T^{2}
47 1+(0.468−8.04i)T+(−46.6−5.45i)T2 1 + (0.468 - 8.04i)T + (-46.6 - 5.45i)T^{2}
53 1+(−3.36−5.83i)T+(−26.5+45.8i)T2 1 + (-3.36 - 5.83i)T + (-26.5 + 45.8i)T^{2}
59 1+(−8.40−0.982i)T+(57.4+13.6i)T2 1 + (-8.40 - 0.982i)T + (57.4 + 13.6i)T^{2}
61 1+(−0.457+0.229i)T+(36.4−48.9i)T2 1 + (-0.457 + 0.229i)T + (36.4 - 48.9i)T^{2}
67 1+(5.52+5.85i)T+(−3.89+66.8i)T2 1 + (5.52 + 5.85i)T + (-3.89 + 66.8i)T^{2}
71 1+(−12.2+10.2i)T+(12.3−69.9i)T2 1 + (-12.2 + 10.2i)T + (12.3 - 69.9i)T^{2}
73 1+(10.6+8.90i)T+(12.6+71.8i)T2 1 + (10.6 + 8.90i)T + (12.6 + 71.8i)T^{2}
79 1+(−15.0−3.56i)T+(70.5+35.4i)T2 1 + (-15.0 - 3.56i)T + (70.5 + 35.4i)T^{2}
83 1+(7.27+1.72i)T+(74.1+37.2i)T2 1 + (7.27 + 1.72i)T + (74.1 + 37.2i)T^{2}
89 1+(−11.9−10.0i)T+(15.4+87.6i)T2 1 + (-11.9 - 10.0i)T + (15.4 + 87.6i)T^{2}
97 1+(9.83+13.2i)T+(−27.8+92.9i)T2 1 + (9.83 + 13.2i)T + (-27.8 + 92.9i)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

−12.76325902971017274827917315852, −11.76831997518847206490585808245, −10.72375237403623160788775920009, −9.405755854401616291273359642971, −8.764053803992048045040668713083, −7.74962537058962419725020606305, −6.85296990276714644681684812963, −5.07918891841439800711107696594, −3.16323106380685883668107451132, −1.77024750012067989821083313495, 2.03934956695346106728769090903, 3.76489310020126840729830417693, 5.26695245496582049391488325955, 6.98437882859923122776251947180, 8.061048998319594800040307786201, 8.641827704773554398570741530554, 10.04896326122633187178464334709, 10.48189804282574824491389037176, 11.74149003639693877213214753275, 13.17344736393031605639211900366

Graph of the ZZ-function along the critical line