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Properties

Label 2-162-81.25-c1-0-7
Degree 22
Conductor 162162
Sign 0.826+0.563i0.826 + 0.563i
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  + (0.893 − 0.448i)2-s + (1.51 − 0.845i)3-s + (0.597 − 0.802i)4-s + (−0.379 + 1.26i)5-s + (0.971 − 1.43i)6-s + (−0.768 + 1.78i)7-s + (0.173 − 0.984i)8-s + (1.57 − 2.55i)9-s + (0.229 + 1.30i)10-s + (−2.10 − 0.499i)11-s + (0.224 − 1.71i)12-s + (−1.69 + 1.11i)13-s + (0.112 + 1.93i)14-s + (0.497 + 2.23i)15-s + (−0.286 − 0.957i)16-s + (−7.02 − 2.55i)17-s + ⋯
L(s)  = 1  + (0.631 − 0.317i)2-s + (0.872 − 0.488i)3-s + (0.298 − 0.401i)4-s + (−0.169 + 0.566i)5-s + (0.396 − 0.585i)6-s + (−0.290 + 0.673i)7-s + (0.0613 − 0.348i)8-s + (0.523 − 0.851i)9-s + (0.0726 + 0.412i)10-s + (−0.635 − 0.150i)11-s + (0.0648 − 0.495i)12-s + (−0.471 + 0.309i)13-s + (0.0301 + 0.517i)14-s + (0.128 + 0.577i)15-s + (−0.0717 − 0.239i)16-s + (−1.70 − 0.619i)17-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.826+0.563i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+1/2) L(s)=((0.826+0.563i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.826+0.563i0.826 + 0.563i
Analytic conductor: 1.293571.29357
Root analytic conductor: 1.137351.13735
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ162(25,⋅)\chi_{162} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :1/2), 0.826+0.563i)(2,\ 162,\ (\ :1/2),\ 0.826 + 0.563i)

Particular Values

L(1)L(1) ≈\approx 1.75815−0.542027i1.75815 - 0.542027i
L(12)L(\frac12) ≈\approx 1.75815−0.542027i1.75815 - 0.542027i
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+(−0.893+0.448i)T 1 + (-0.893 + 0.448i)T
3 1+(−1.51+0.845i)T 1 + (-1.51 + 0.845i)T
good5 1+(0.379−1.26i)T+(−4.17−2.74i)T2 1 + (0.379 - 1.26i)T + (-4.17 - 2.74i)T^{2}
7 1+(0.768−1.78i)T+(−4.80−5.09i)T2 1 + (0.768 - 1.78i)T + (-4.80 - 5.09i)T^{2}
11 1+(2.10+0.499i)T+(9.82+4.93i)T2 1 + (2.10 + 0.499i)T + (9.82 + 4.93i)T^{2}
13 1+(1.69−1.11i)T+(5.14−11.9i)T2 1 + (1.69 - 1.11i)T + (5.14 - 11.9i)T^{2}
17 1+(7.02+2.55i)T+(13.0+10.9i)T2 1 + (7.02 + 2.55i)T + (13.0 + 10.9i)T^{2}
19 1+(−2.13+0.776i)T+(14.5−12.2i)T2 1 + (-2.13 + 0.776i)T + (14.5 - 12.2i)T^{2}
23 1+(−1.34−3.12i)T+(−15.7+16.7i)T2 1 + (-1.34 - 3.12i)T + (-15.7 + 16.7i)T^{2}
29 1+(0.282−4.85i)T+(−28.8−3.36i)T2 1 + (0.282 - 4.85i)T + (-28.8 - 3.36i)T^{2}
31 1+(−5.07−0.593i)T+(30.1+7.14i)T2 1 + (-5.07 - 0.593i)T + (30.1 + 7.14i)T^{2}
37 1+(−4.70+3.95i)T+(6.42−36.4i)T2 1 + (-4.70 + 3.95i)T + (6.42 - 36.4i)T^{2}
41 1+(6.98+3.50i)T+(24.4+32.8i)T2 1 + (6.98 + 3.50i)T + (24.4 + 32.8i)T^{2}
43 1+(4.77−5.06i)T+(−2.50−42.9i)T2 1 + (4.77 - 5.06i)T + (-2.50 - 42.9i)T^{2}
47 1+(−4.18+0.489i)T+(45.7−10.8i)T2 1 + (-4.18 + 0.489i)T + (45.7 - 10.8i)T^{2}
53 1+(5.39+9.34i)T+(−26.5+45.8i)T2 1 + (5.39 + 9.34i)T + (-26.5 + 45.8i)T^{2}
59 1+(−6.72+1.59i)T+(52.7−26.4i)T2 1 + (-6.72 + 1.59i)T + (52.7 - 26.4i)T^{2}
61 1+(5.37+7.22i)T+(−17.4+58.4i)T2 1 + (5.37 + 7.22i)T + (-17.4 + 58.4i)T^{2}
67 1+(−0.661−11.3i)T+(−66.5+7.77i)T2 1 + (-0.661 - 11.3i)T + (-66.5 + 7.77i)T^{2}
71 1+(−0.518−2.93i)T+(−66.7+24.2i)T2 1 + (-0.518 - 2.93i)T + (-66.7 + 24.2i)T^{2}
73 1+(−1.80+10.2i)T+(−68.5−24.9i)T2 1 + (-1.80 + 10.2i)T + (-68.5 - 24.9i)T^{2}
79 1+(−9.16+4.60i)T+(47.1−63.3i)T2 1 + (-9.16 + 4.60i)T + (47.1 - 63.3i)T^{2}
83 1+(−3.24+1.62i)T+(49.5−66.5i)T2 1 + (-3.24 + 1.62i)T + (49.5 - 66.5i)T^{2}
89 1+(−1.42+8.10i)T+(−83.6−30.4i)T2 1 + (-1.42 + 8.10i)T + (-83.6 - 30.4i)T^{2}
97 1+(2.30+7.70i)T+(−81.0+53.3i)T2 1 + (2.30 + 7.70i)T + (-81.0 + 53.3i)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

−12.97451261709878170533866314158, −11.92648594114697248770163198876, −11.00164397893555184481316029717, −9.662224409704637151712914945783, −8.748880957263855169509662424950, −7.33634388920758710470151894067, −6.51779014768770218148427810369, −4.92133680838536423851680710945, −3.26663303651104179350910083746, −2.33405441117148183421818350509, 2.63315780385185401385809354712, 4.13645821561245840735979459930, 4.91625471803565595118624520757, 6.65072608839464935183486837356, 7.84755644307340192056579291835, 8.667280371216183278436065813415, 9.948613477604685201377437716990, 10.84794429737321200813015794386, 12.30754034306689090318410998372, 13.31077528081622393229804198323

Graph of the ZZ-function along the critical line