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Properties

Label 2-162-9.7-c7-0-3
Degree 22
Conductor 162162
Sign −0.642+0.766i-0.642 + 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 + (−5.62 − 9.73i)5-s + (−719. + 1.24e3i)7-s + 511.·8-s + 89.9·10-s + (−2.21e3 + 3.83e3i)11-s + (3.61e3 + 6.26e3i)13-s + (−5.75e3 − 9.97e3i)14-s + (−2.04e3 + 3.54e3i)16-s + 1.32e4·17-s + 1.35e4·19-s + (−359. + 623. i)20-s + (−1.77e4 − 3.07e4i)22-s + (3.28e4 + 5.69e4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0201 − 0.0348i)5-s + (−0.793 + 1.37i)7-s + 0.353·8-s + 0.0284·10-s + (−0.502 + 0.869i)11-s + (0.456 + 0.790i)13-s + (−0.560 − 0.971i)14-s + (−0.125 + 0.216i)16-s + 0.652·17-s + 0.451·19-s + (−0.0100 + 0.0174i)20-s + (−0.355 − 0.614i)22-s + (0.563 + 0.975i)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+0.766i-0.642 + 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 0.60536256430.6053625643
L(12)L(\frac12) ≈\approx 0.60536256430.6053625643
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+(5.62+9.73i)T+(−3.90e4+6.76e4i)T2 1 + (5.62 + 9.73i)T + (-3.90e4 + 6.76e4i)T^{2}
7 1+(719.−1.24e3i)T+(−4.11e5−7.13e5i)T2 1 + (719. - 1.24e3i)T + (-4.11e5 - 7.13e5i)T^{2}
11 1+(2.21e3−3.83e3i)T+(−9.74e6−1.68e7i)T2 1 + (2.21e3 - 3.83e3i)T + (-9.74e6 - 1.68e7i)T^{2}
13 1+(−3.61e3−6.26e3i)T+(−3.13e7+5.43e7i)T2 1 + (-3.61e3 - 6.26e3i)T + (-3.13e7 + 5.43e7i)T^{2}
17 1−1.32e4T+4.10e8T2 1 - 1.32e4T + 4.10e8T^{2}
19 1−1.35e4T+8.93e8T2 1 - 1.35e4T + 8.93e8T^{2}
23 1+(−3.28e4−5.69e4i)T+(−1.70e9+2.94e9i)T2 1 + (-3.28e4 - 5.69e4i)T + (-1.70e9 + 2.94e9i)T^{2}
29 1+(7.42e4−1.28e5i)T+(−8.62e9−1.49e10i)T2 1 + (7.42e4 - 1.28e5i)T + (-8.62e9 - 1.49e10i)T^{2}
31 1+(6.94e4+1.20e5i)T+(−1.37e10+2.38e10i)T2 1 + (6.94e4 + 1.20e5i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+1.21e5T+9.49e10T2 1 + 1.21e5T + 9.49e10T^{2}
41 1+(2.63e4+4.57e4i)T+(−9.73e10+1.68e11i)T2 1 + (2.63e4 + 4.57e4i)T + (-9.73e10 + 1.68e11i)T^{2}
43 1+(3.75e5−6.51e5i)T+(−1.35e11−2.35e11i)T2 1 + (3.75e5 - 6.51e5i)T + (-1.35e11 - 2.35e11i)T^{2}
47 1+(4.91e5−8.51e5i)T+(−2.53e11−4.38e11i)T2 1 + (4.91e5 - 8.51e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+6.21e5T+1.17e12T2 1 + 6.21e5T + 1.17e12T^{2}
59 1+(4.07e5+7.05e5i)T+(−1.24e12+2.15e12i)T2 1 + (4.07e5 + 7.05e5i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(−1.23e6+2.13e6i)T+(−1.57e12−2.72e12i)T2 1 + (-1.23e6 + 2.13e6i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(1.72e6+2.99e6i)T+(−3.03e12+5.24e12i)T2 1 + (1.72e6 + 2.99e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1−3.64e6T+9.09e12T2 1 - 3.64e6T + 9.09e12T^{2}
73 1+5.55e6T+1.10e13T2 1 + 5.55e6T + 1.10e13T^{2}
79 1+(−1.02e6+1.77e6i)T+(−9.60e12−1.66e13i)T2 1 + (-1.02e6 + 1.77e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+(5.13e6−8.88e6i)T+(−1.35e13−2.35e13i)T2 1 + (5.13e6 - 8.88e6i)T + (-1.35e13 - 2.35e13i)T^{2}
89 1−8.12e6T+4.42e13T2 1 - 8.12e6T + 4.42e13T^{2}
97 1+(−6.64e6+1.15e7i)T+(−4.03e13−6.99e13i)T2 1 + (-6.64e6 + 1.15e7i)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

−12.31039768106890922110017061689, −11.16952891818876009100888264938, −9.719387971440982894210454510478, −9.252447415816972584987761916363, −8.094527599978137771758316888382, −6.92417885483153803624900046771, −5.91066637311018095030120865747, −4.90789543430469953064019459919, −3.15281217337257346854871928542, −1.69387552978732328603835821663, 0.20780042925037711965450096921, 1.08305847857385341894271348662, 3.02872539211603706842819759593, 3.75569407739017611735964207410, 5.38786971024530444399090246530, 6.85603280902836257994594323136, 7.85179428645562890058274313293, 8.956564618606019168333674198703, 10.28629193658380531326467219819, 10.57276827953500966588498236863

Graph of the ZZ-function along the critical line