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Properties

Label 2-162-81.76-c1-0-5
Degree 22
Conductor 162162
Sign 0.984+0.177i0.984 + 0.177i
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 + (1.70 − 0.290i)3-s + (0.893 − 0.448i)4-s + (0.537 + 0.721i)5-s + (1.59 − 0.676i)6-s + (−3.95 + 2.60i)7-s + (0.766 − 0.642i)8-s + (2.83 − 0.992i)9-s + (0.689 + 0.578i)10-s + (−4.21 − 0.492i)11-s + (1.39 − 1.02i)12-s + (−1.75 − 5.86i)13-s + (−3.24 + 3.44i)14-s + (1.12 + 1.07i)15-s + (0.597 − 0.802i)16-s + (−0.432 + 2.45i)17-s + ⋯
L(s)  = 1  + (0.688 − 0.163i)2-s + (0.985 − 0.167i)3-s + (0.446 − 0.224i)4-s + (0.240 + 0.322i)5-s + (0.650 − 0.276i)6-s + (−1.49 + 0.983i)7-s + (0.270 − 0.227i)8-s + (0.943 − 0.330i)9-s + (0.217 + 0.182i)10-s + (−1.26 − 0.148i)11-s + (0.402 − 0.296i)12-s + (−0.487 − 1.62i)13-s + (−0.868 + 0.920i)14-s + (0.291 + 0.277i)15-s + (0.149 − 0.200i)16-s + (−0.104 + 0.594i)17-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.984+0.177i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+1/2) L(s)=((0.984+0.177i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.984+0.177i0.984 + 0.177i
Analytic conductor: 1.293571.29357
Root analytic conductor: 1.137351.13735
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ162(157,⋅)\chi_{162} (157, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :1/2), 0.984+0.177i)(2,\ 162,\ (\ :1/2),\ 0.984 + 0.177i)

Particular Values

L(1)L(1) ≈\approx 1.91919−0.171820i1.91919 - 0.171820i
L(12)L(\frac12) ≈\approx 1.91919−0.171820i1.91919 - 0.171820i
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+(−1.70+0.290i)T 1 + (-1.70 + 0.290i)T
good5 1+(−0.537−0.721i)T+(−1.43+4.78i)T2 1 + (-0.537 - 0.721i)T + (-1.43 + 4.78i)T^{2}
7 1+(3.95−2.60i)T+(2.77−6.42i)T2 1 + (3.95 - 2.60i)T + (2.77 - 6.42i)T^{2}
11 1+(4.21+0.492i)T+(10.7+2.53i)T2 1 + (4.21 + 0.492i)T + (10.7 + 2.53i)T^{2}
13 1+(1.75+5.86i)T+(−10.8+7.14i)T2 1 + (1.75 + 5.86i)T + (-10.8 + 7.14i)T^{2}
17 1+(0.432−2.45i)T+(−15.9−5.81i)T2 1 + (0.432 - 2.45i)T + (-15.9 - 5.81i)T^{2}
19 1+(0.284+1.61i)T+(−17.8+6.49i)T2 1 + (0.284 + 1.61i)T + (-17.8 + 6.49i)T^{2}
23 1+(−6.35−4.17i)T+(9.10+21.1i)T2 1 + (-6.35 - 4.17i)T + (9.10 + 21.1i)T^{2}
29 1+(−2.09−2.21i)T+(−1.68+28.9i)T2 1 + (-2.09 - 2.21i)T + (-1.68 + 28.9i)T^{2}
31 1+(0.107−1.84i)T+(−30.7−3.59i)T2 1 + (0.107 - 1.84i)T + (-30.7 - 3.59i)T^{2}
37 1+(8.42−3.06i)T+(28.3−23.7i)T2 1 + (8.42 - 3.06i)T + (28.3 - 23.7i)T^{2}
41 1+(−5.04−1.19i)T+(36.6+18.4i)T2 1 + (-5.04 - 1.19i)T + (36.6 + 18.4i)T^{2}
43 1+(−1.06−2.46i)T+(−29.5+31.2i)T2 1 + (-1.06 - 2.46i)T + (-29.5 + 31.2i)T^{2}
47 1+(−0.486−8.35i)T+(−46.6+5.45i)T2 1 + (-0.486 - 8.35i)T + (-46.6 + 5.45i)T^{2}
53 1+(1.11−1.93i)T+(−26.5−45.8i)T2 1 + (1.11 - 1.93i)T + (-26.5 - 45.8i)T^{2}
59 1+(3.71−0.433i)T+(57.4−13.6i)T2 1 + (3.71 - 0.433i)T + (57.4 - 13.6i)T^{2}
61 1+(−2.81−1.41i)T+(36.4+48.9i)T2 1 + (-2.81 - 1.41i)T + (36.4 + 48.9i)T^{2}
67 1+(−3.28+3.48i)T+(−3.89−66.8i)T2 1 + (-3.28 + 3.48i)T + (-3.89 - 66.8i)T^{2}
71 1+(3.18+2.67i)T+(12.3+69.9i)T2 1 + (3.18 + 2.67i)T + (12.3 + 69.9i)T^{2}
73 1+(−1.09+0.922i)T+(12.6−71.8i)T2 1 + (-1.09 + 0.922i)T + (12.6 - 71.8i)T^{2}
79 1+(5.03−1.19i)T+(70.5−35.4i)T2 1 + (5.03 - 1.19i)T + (70.5 - 35.4i)T^{2}
83 1+(2.63−0.624i)T+(74.1−37.2i)T2 1 + (2.63 - 0.624i)T + (74.1 - 37.2i)T^{2}
89 1+(−12.5+10.5i)T+(15.4−87.6i)T2 1 + (-12.5 + 10.5i)T + (15.4 - 87.6i)T^{2}
97 1+(−2.79+3.75i)T+(−27.8−92.9i)T2 1 + (-2.79 + 3.75i)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

−12.79463850023032950976361596949, −12.52773176027738623641992322486, −10.63531137735042718010294738982, −9.922171153356912434171272250920, −8.777445811289761209946368885212, −7.54684454753043927427250536867, −6.34470074635749085602795074163, −5.20912162212214439184039484586, −3.10460446898260012657750010895, −2.76745412678928797247351510067, 2.54352155780279284700628637825, 3.80400254564471347820740126459, 4.95543185954429229297530967145, 6.76930066405254153411832754788, 7.35911102685172985824918700081, 8.923740243562694930650066007403, 9.802519275269370745863797597508, 10.71897108539107005664582091102, 12.39165404853972178686072889063, 13.19134611871119355281362995050

Graph of the ZZ-function along the critical line