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Properties

Label 2-162-9.7-c7-0-5
Degree 22
Conductor 162162
Sign 0.642−0.766i0.642 - 0.766i
Analytic cond. 50.606350.6063
Root an. cond. 7.113817.11381
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (−235. − 408. i)5-s + (−367. + 636. i)7-s + 511.·8-s + 3.77e3·10-s + (−2.66e3 + 4.60e3i)11-s + (−4.43e3 − 7.67e3i)13-s + (−2.94e3 − 5.09e3i)14-s + (−2.04e3 + 3.54e3i)16-s − 3.41e4·17-s − 8.70e3·19-s + (−1.50e4 + 2.61e4i)20-s + (−2.12e4 − 3.68e4i)22-s + (−2.60e4 − 4.51e4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.843 − 1.46i)5-s + (−0.405 + 0.701i)7-s + 0.353·8-s + 1.19·10-s + (−0.602 + 1.04i)11-s + (−0.559 − 0.969i)13-s + (−0.286 − 0.496i)14-s + (−0.125 + 0.216i)16-s − 1.68·17-s − 0.291·19-s + (−0.421 + 0.730i)20-s + (−0.426 − 0.738i)22-s + (−0.446 − 0.773i)23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.642−0.766i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+7/2) L(s)=((0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.642−0.766i0.642 - 0.766i
Analytic conductor: 50.606350.6063
Root analytic conductor: 7.113817.11381
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ162(55,⋅)\chi_{162} (55, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :7/2), 0.642−0.766i)(2,\ 162,\ (\ :7/2),\ 0.642 - 0.766i)

Particular Values

L(4)L(4) ≈\approx 0.56073546040.5607354604
L(12)L(\frac12) ≈\approx 0.56073546040.5607354604
L(92)L(\frac{9}{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+(4−6.92i)T 1 + (4 - 6.92i)T
3 1 1
good5 1+(235.+408.i)T+(−3.90e4+6.76e4i)T2 1 + (235. + 408. i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(367.−636.i)T+(−4.11e5−7.13e5i)T2 1 + (367. - 636. i)T + (-4.11e5 - 7.13e5i)T^{2}
11 1+(2.66e3−4.60e3i)T+(−9.74e6−1.68e7i)T2 1 + (2.66e3 - 4.60e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(4.43e3+7.67e3i)T+(−3.13e7+5.43e7i)T2 1 + (4.43e3 + 7.67e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1+3.41e4T+4.10e8T2 1 + 3.41e4T + 4.10e8T^{2}
19 1+8.70e3T+8.93e8T2 1 + 8.70e3T + 8.93e8T^{2}
23 1+(2.60e4+4.51e4i)T+(−1.70e9+2.94e9i)T2 1 + (2.60e4 + 4.51e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(−1.06e5+1.84e5i)T+(−8.62e9−1.49e10i)T2 1 + (-1.06e5 + 1.84e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+(−1.16e5−2.01e5i)T+(−1.37e10+2.38e10i)T2 1 + (-1.16e5 - 2.01e5i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1−6.73e4T+9.49e10T2 1 - 6.73e4T + 9.49e10T^{2}
41 1+(−2.25e4−3.90e4i)T+(−9.73e10+1.68e11i)T2 1 + (-2.25e4 - 3.90e4i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+(1.07e5−1.85e5i)T+(−1.35e11−2.35e11i)T2 1 + (1.07e5 - 1.85e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+(−2.80e5+4.85e5i)T+(−2.53e11−4.38e11i)T2 1 + (-2.80e5 + 4.85e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+2.44e5T+1.17e12T2 1 + 2.44e5T + 1.17e12T^{2}
59 1+(9.66e5+1.67e6i)T+(−1.24e12+2.15e12i)T2 1 + (9.66e5 + 1.67e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.06e6−1.84e6i)T+(−1.57e12−2.72e12i)T2 1 + (1.06e6 - 1.84e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(−8.14e5−1.41e6i)T+(−3.03e12+5.24e12i)T2 1 + (-8.14e5 - 1.41e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1−5.38e6T+9.09e12T2 1 - 5.38e6T + 9.09e12T^{2}
73 1+1.01e6T+1.10e13T2 1 + 1.01e6T + 1.10e13T^{2}
79 1+(−3.87e6+6.70e6i)T+(−9.60e12−1.66e13i)T2 1 + (-3.87e6 + 6.70e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+(1.04e6−1.81e6i)T+(−1.35e13−2.35e13i)T2 1 + (1.04e6 - 1.81e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1−1.17e7T+4.42e13T2 1 - 1.17e7T + 4.42e13T^{2}
97 1+(9.61e5−1.66e6i)T+(−4.03e13−6.99e13i)T2 1 + (9.61e5 - 1.66e6i)T + (-4.03e13 - 6.99e13i)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

−12.00623627525008745674351045109, −10.45870156078439090842471292309, −9.370889438239395012661086394119, −8.484531124864597373201688008136, −7.80761514221843975512472991126, −6.45112814590231201236759327417, −5.05164377956970977674208037733, −4.42268968357139324854598370906, −2.34479546983453816364319512662, −0.55395323705862797919098087384, 0.31853878932003977321339357356, 2.34117534416299320622998616371, 3.36503305745231424375723216326, 4.35211546290866933134079740307, 6.45828164291577771189593479498, 7.23008229708142280760882464389, 8.291257550547796394814422162652, 9.586248279992620643019574331470, 10.80230844067287965269215245834, 11.03540444856480751740925215503

Graph of the ZZ-function along the critical line