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Properties

Label 2-162-27.4-c1-0-1
Degree 22
Conductor 162162
Sign 0.557−0.830i0.557 - 0.830i
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.617 + 3.49i)5-s + (−0.244 + 0.205i)7-s + (0.500 + 0.866i)8-s + (−1.77 + 3.07i)10-s + (−0.773 − 4.38i)11-s + (4.39 − 1.60i)13-s + (−0.300 + 0.109i)14-s + (0.173 + 0.984i)16-s + (0.567 − 0.982i)17-s + (−0.928 − 1.60i)19-s + (−2.72 + 2.28i)20-s + (0.773 − 4.38i)22-s + (−0.110 − 0.0926i)23-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.383 + 0.321i)4-s + (−0.275 + 1.56i)5-s + (−0.0925 + 0.0776i)7-s + (0.176 + 0.306i)8-s + (−0.561 + 0.973i)10-s + (−0.233 − 1.32i)11-s + (1.21 − 0.443i)13-s + (−0.0802 + 0.0292i)14-s + (0.0434 + 0.246i)16-s + (0.137 − 0.238i)17-s + (−0.213 − 0.369i)19-s + (−0.608 + 0.510i)20-s + (0.164 − 0.934i)22-s + (−0.0230 − 0.0193i)23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.557−0.830i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+1/2) L(s)=((0.557−0.830i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.557−0.830i0.557 - 0.830i
Analytic conductor: 1.293571.29357
Root analytic conductor: 1.137351.13735
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ162(145,⋅)\chi_{162} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :1/2), 0.557−0.830i)(2,\ 162,\ (\ :1/2),\ 0.557 - 0.830i)

Particular Values

L(1)L(1) ≈\approx 1.37040+0.730359i1.37040 + 0.730359i
L(12)L(\frac12) ≈\approx 1.37040+0.730359i1.37040 + 0.730359i
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+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
3 1 1
good5 1+(0.617−3.49i)T+(−4.69−1.71i)T2 1 + (0.617 - 3.49i)T + (-4.69 - 1.71i)T^{2}
7 1+(0.244−0.205i)T+(1.21−6.89i)T2 1 + (0.244 - 0.205i)T + (1.21 - 6.89i)T^{2}
11 1+(0.773+4.38i)T+(−10.3+3.76i)T2 1 + (0.773 + 4.38i)T + (-10.3 + 3.76i)T^{2}
13 1+(−4.39+1.60i)T+(9.95−8.35i)T2 1 + (-4.39 + 1.60i)T + (9.95 - 8.35i)T^{2}
17 1+(−0.567+0.982i)T+(−8.5−14.7i)T2 1 + (-0.567 + 0.982i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.928+1.60i)T+(−9.5+16.4i)T2 1 + (0.928 + 1.60i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.110+0.0926i)T+(3.99+22.6i)T2 1 + (0.110 + 0.0926i)T + (3.99 + 22.6i)T^{2}
29 1+(4.09+1.49i)T+(22.2+18.6i)T2 1 + (4.09 + 1.49i)T + (22.2 + 18.6i)T^{2}
31 1+(−0.514−0.431i)T+(5.38+30.5i)T2 1 + (-0.514 - 0.431i)T + (5.38 + 30.5i)T^{2}
37 1+(3.79−6.57i)T+(−18.5−32.0i)T2 1 + (3.79 - 6.57i)T + (-18.5 - 32.0i)T^{2}
41 1+(−2.04+0.744i)T+(31.4−26.3i)T2 1 + (-2.04 + 0.744i)T + (31.4 - 26.3i)T^{2}
43 1+(1.23+6.98i)T+(−40.4+14.7i)T2 1 + (1.23 + 6.98i)T + (-40.4 + 14.7i)T^{2}
47 1+(−7.91+6.63i)T+(8.16−46.2i)T2 1 + (-7.91 + 6.63i)T + (8.16 - 46.2i)T^{2}
53 1+0.805T+53T2 1 + 0.805T + 53T^{2}
59 1+(0.517−2.93i)T+(−55.4−20.1i)T2 1 + (0.517 - 2.93i)T + (-55.4 - 20.1i)T^{2}
61 1+(2.67−2.24i)T+(10.5−60.0i)T2 1 + (2.67 - 2.24i)T + (10.5 - 60.0i)T^{2}
67 1+(6.99−2.54i)T+(51.3−43.0i)T2 1 + (6.99 - 2.54i)T + (51.3 - 43.0i)T^{2}
71 1+(4.04−7.01i)T+(−35.5−61.4i)T2 1 + (4.04 - 7.01i)T + (-35.5 - 61.4i)T^{2}
73 1+(7.30+12.6i)T+(−36.5+63.2i)T2 1 + (7.30 + 12.6i)T + (-36.5 + 63.2i)T^{2}
79 1+(−11.8−4.30i)T+(60.5+50.7i)T2 1 + (-11.8 - 4.30i)T + (60.5 + 50.7i)T^{2}
83 1+(5.08+1.85i)T+(63.5+53.3i)T2 1 + (5.08 + 1.85i)T + (63.5 + 53.3i)T^{2}
89 1+(−2.52−4.37i)T+(−44.5+77.0i)T2 1 + (-2.52 - 4.37i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.24−18.3i)T+(−91.1+33.1i)T2 1 + (-3.24 - 18.3i)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

−13.42648921850884539623893944856, −11.91067949518287788232450594121, −11.01568360746577456062406856720, −10.49005561460321290454973232752, −8.711100055481158769212833980497, −7.58430899762155296303026052096, −6.49893976630276589747969973201, −5.67934625882443467408105351919, −3.73620688801197560256892678594, −2.88472112685116412134947135781, 1.63739275258173341701393103055, 3.90995933279111134493845871754, 4.78861018298391905678126954744, 5.96625878960616884528932471475, 7.47814765900931310930926018958, 8.679586147252383293036129879272, 9.637783624729829551495430835392, 10.91306525939837882299533472502, 12.06163130308780512316081627601, 12.71466121373404654439906401870

Graph of the ZZ-function along the critical line