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Properties

Label 2-162-9.4-c7-0-25
Degree $2$
Conductor $162$
Sign $0.173 + 0.984i$
Analytic cond. $50.6063$
Root an. cond. $7.11381$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(50.6063\)
Root analytic conductor: \(7.11381\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :7/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.131507544\)
\(L(\frac12)\) \(\approx\) \(2.131507544\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 6.92i)T \)
3 \( 1 \)
good5 \( 1 + (-232. + 403. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-24.8 - 43.0i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (151. + 262. i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-2.75e3 + 4.77e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 2.26e4T + 4.10e8T^{2} \)
19 \( 1 - 5.28e4T + 8.93e8T^{2} \)
23 \( 1 + (1.00e4 - 1.73e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (1.25e5 + 2.16e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (1.31e5 - 2.27e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + 4.60e5T + 9.49e10T^{2} \)
41 \( 1 + (-1.20e5 + 2.08e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (2.84e5 + 4.92e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (4.37e5 + 7.57e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 - 8.11e5T + 1.17e12T^{2} \)
59 \( 1 + (-6.46e5 + 1.11e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-8.14e5 - 1.41e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-9.21e5 + 1.59e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 4.56e6T + 9.09e12T^{2} \)
73 \( 1 + 1.91e6T + 1.10e13T^{2} \)
79 \( 1 + (-9.05e5 - 1.56e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.54e6 + 2.67e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 4.10e6T + 4.42e13T^{2} \)
97 \( 1 + (3.95e6 + 6.85e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line