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Properties

Label 2-162-9.4-c7-0-25
Degree 22
Conductor 162162
Sign 0.173+0.984i0.173 + 0.984i
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 + (232. − 403. i)5-s + (24.8 + 43.0i)7-s − 511.·8-s + 3.72e3·10-s + (−151. − 262. i)11-s + (2.75e3 − 4.77e3i)13-s + (−198. + 344. i)14-s + (−2.04e3 − 3.54e3i)16-s − 2.26e4·17-s + 5.28e4·19-s + (1.49e4 + 2.58e4i)20-s + (1.21e3 − 2.09e3i)22-s + (−1.00e4 + 1.73e4i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.833 − 1.44i)5-s + (0.0274 + 0.0474i)7-s − 0.353·8-s + 1.17·10-s + (−0.0343 − 0.0594i)11-s + (0.348 − 0.602i)13-s + (−0.0193 + 0.0335i)14-s + (−0.125 − 0.216i)16-s − 1.11·17-s + 1.76·19-s + (0.416 + 0.721i)20-s + (0.0242 − 0.0420i)22-s + (−0.171 + 0.297i)23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+7/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.173+0.984i0.173 + 0.984i
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), 0.173+0.984i)(2,\ 162,\ (\ :7/2),\ 0.173 + 0.984i)

Particular Values

L(4)L(4) ≈\approx 2.1315075442.131507544
L(12)L(\frac12) ≈\approx 2.1315075442.131507544
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+(−232.+403.i)T+(−3.90e4−6.76e4i)T2 1 + (-232. + 403. i)T + (-3.90e4 - 6.76e4i)T^{2}
7 1+(−24.8−43.0i)T+(−4.11e5+7.13e5i)T2 1 + (-24.8 - 43.0i)T + (-4.11e5 + 7.13e5i)T^{2}
11 1+(151.+262.i)T+(−9.74e6+1.68e7i)T2 1 + (151. + 262. i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−2.75e3+4.77e3i)T+(−3.13e7−5.43e7i)T2 1 + (-2.75e3 + 4.77e3i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1+2.26e4T+4.10e8T2 1 + 2.26e4T + 4.10e8T^{2}
19 1−5.28e4T+8.93e8T2 1 - 5.28e4T + 8.93e8T^{2}
23 1+(1.00e4−1.73e4i)T+(−1.70e9−2.94e9i)T2 1 + (1.00e4 - 1.73e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(1.25e5+2.16e5i)T+(−8.62e9+1.49e10i)T2 1 + (1.25e5 + 2.16e5i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+(1.31e5−2.27e5i)T+(−1.37e10−2.38e10i)T2 1 + (1.31e5 - 2.27e5i)T + (-1.37e10 - 2.38e10i)T^{2}
37 1+4.60e5T+9.49e10T2 1 + 4.60e5T + 9.49e10T^{2}
41 1+(−1.20e5+2.08e5i)T+(−9.73e10−1.68e11i)T2 1 + (-1.20e5 + 2.08e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(2.84e5+4.92e5i)T+(−1.35e11+2.35e11i)T2 1 + (2.84e5 + 4.92e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+(4.37e5+7.57e5i)T+(−2.53e11+4.38e11i)T2 1 + (4.37e5 + 7.57e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1−8.11e5T+1.17e12T2 1 - 8.11e5T + 1.17e12T^{2}
59 1+(−6.46e5+1.11e6i)T+(−1.24e12−2.15e12i)T2 1 + (-6.46e5 + 1.11e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−8.14e5−1.41e6i)T+(−1.57e12+2.72e12i)T2 1 + (-8.14e5 - 1.41e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(−9.21e5+1.59e6i)T+(−3.03e12−5.24e12i)T2 1 + (-9.21e5 + 1.59e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1−4.56e6T+9.09e12T2 1 - 4.56e6T + 9.09e12T^{2}
73 1+1.91e6T+1.10e13T2 1 + 1.91e6T + 1.10e13T^{2}
79 1+(−9.05e5−1.56e6i)T+(−9.60e12+1.66e13i)T2 1 + (-9.05e5 - 1.56e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(1.54e6+2.67e6i)T+(−1.35e13+2.35e13i)T2 1 + (1.54e6 + 2.67e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1+4.10e6T+4.42e13T2 1 + 4.10e6T + 4.42e13T^{2}
97 1+(3.95e6+6.85e6i)T+(−4.03e13+6.99e13i)T2 1 + (3.95e6 + 6.85e6i)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

−11.60634035391090622644214396243, −10.04567303086214086062146217172, −9.069762590116572012713573502214, −8.345956162253071006461662887805, −7.03414948686386863564285974244, −5.58178123144890817411560020757, −5.17317954449258284660878269553, −3.71865726139998912413924850373, −1.88358271253472456898494474782, −0.47440866646466397808550475406, 1.57480960757277150628392523661, 2.65420870713220137846497856502, 3.71699923669772855884839863412, 5.31007376486406484158483537698, 6.42038954446723147936844326471, 7.32150513728083552144010434513, 9.119113201911436154747873160466, 9.908058795167725267443037712660, 10.97122554799296886477292211578, 11.42796803012033589142562193227

Graph of the ZZ-function along the critical line