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Properties

Label 2-162-9.4-c7-0-16
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 + (140. − 242. i)5-s + (82.1 + 142. i)7-s + 511.·8-s − 2.24e3·10-s + (1.01e3 + 1.76e3i)11-s + (839. − 1.45e3i)13-s + (657. − 1.13e3i)14-s + (−2.04e3 − 3.54e3i)16-s + 3.16e4·17-s − 1.26e4·19-s + (8.97e3 + 1.55e4i)20-s + (8.15e3 − 1.41e4i)22-s + (−2.47e4 + 4.28e4i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.501 − 0.868i)5-s + (0.0905 + 0.156i)7-s + 0.353·8-s − 0.709·10-s + (0.230 + 0.399i)11-s + (0.105 − 0.183i)13-s + (0.0640 − 0.110i)14-s + (−0.125 − 0.216i)16-s + 1.56·17-s − 0.422·19-s + (0.250 + 0.434i)20-s + (0.163 − 0.282i)22-s + (−0.423 + 0.734i)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(109,⋅)\chi_{162} (109, \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 1.9683506841.968350684
L(12)L(\frac12) ≈\approx 1.9683506841.968350684
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+(−140.+242.i)T+(−3.90e4−6.76e4i)T2 1 + (-140. + 242. i)T + (-3.90e4 - 6.76e4i)T^{2}
7 1+(−82.1−142.i)T+(−4.11e5+7.13e5i)T2 1 + (-82.1 - 142. i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1+(−1.01e3−1.76e3i)T+(−9.74e6+1.68e7i)T2 1 + (-1.01e3 - 1.76e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−839.+1.45e3i)T+(−3.13e7−5.43e7i)T2 1 + (-839. + 1.45e3i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1−3.16e4T+4.10e8T2 1 - 3.16e4T + 4.10e8T^{2}
19 1+1.26e4T+8.93e8T2 1 + 1.26e4T + 8.93e8T^{2}
23 1+(2.47e4−4.28e4i)T+(−1.70e9−2.94e9i)T2 1 + (2.47e4 - 4.28e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(−3.77e3−6.54e3i)T+(−8.62e9+1.49e10i)T2 1 + (-3.77e3 - 6.54e3i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+(7.80e4−1.35e5i)T+(−1.37e10−2.38e10i)T2 1 + (7.80e4 - 1.35e5i)T + (-1.37e10 - 2.38e10i)T^{2}
37 1−5.41e5T+9.49e10T2 1 - 5.41e5T + 9.49e10T^{2}
41 1+(2.68e5−4.65e5i)T+(−9.73e10−1.68e11i)T2 1 + (2.68e5 - 4.65e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(1.00e5+1.73e5i)T+(−1.35e11+2.35e11i)T2 1 + (1.00e5 + 1.73e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+(−2.05e5−3.55e5i)T+(−2.53e11+4.38e11i)T2 1 + (-2.05e5 - 3.55e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1−1.36e6T+1.17e12T2 1 - 1.36e6T + 1.17e12T^{2}
59 1+(3.99e5−6.91e5i)T+(−1.24e12−2.15e12i)T2 1 + (3.99e5 - 6.91e5i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(2.84e5+4.93e5i)T+(−1.57e12+2.72e12i)T2 1 + (2.84e5 + 4.93e5i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−2.40e6+4.16e6i)T+(−3.03e12−5.24e12i)T2 1 + (-2.40e6 + 4.16e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1−2.45e6T+9.09e12T2 1 - 2.45e6T + 9.09e12T^{2}
73 1−1.60e6T+1.10e13T2 1 - 1.60e6T + 1.10e13T^{2}
79 1+(2.79e6+4.84e6i)T+(−9.60e12+1.66e13i)T2 1 + (2.79e6 + 4.84e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(4.91e6+8.51e6i)T+(−1.35e13+2.35e13i)T2 1 + (4.91e6 + 8.51e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1+1.17e5T+4.42e13T2 1 + 1.17e5T + 4.42e13T^{2}
97 1+(−3.89e6−6.74e6i)T+(−4.03e13+6.99e13i)T2 1 + (-3.89e6 - 6.74e6i)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

−11.49537873698681884216279868592, −10.21016067831766051075176266341, −9.494854408649294685338209683454, −8.538098451118813473306618903313, −7.49981352446716050057523340776, −5.86316711684121368304054443434, −4.79563685059621683008856552041, −3.39659163818577323590618674441, −1.85422764342332677160315632962, −0.880380918656767767778154988007, 0.844046379494394512789461709819, 2.43832735664410802550404803762, 3.95166585076624984543250409010, 5.58362266209070252081425321293, 6.41601462978018089508413745826, 7.46278997314300503246859538317, 8.500872525920530851113755911346, 9.745599058591567801855386568331, 10.44244158922859724776830215172, 11.48802809862510713726347586445

Graph of the ZZ-function along the critical line