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Properties

Label 2-162-9.7-c3-0-7
Degree $2$
Conductor $162$
Sign $-0.173 + 0.984i$
Analytic cond. $9.55830$
Root an. cond. $3.09165$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−1.5 − 2.59i)5-s + (−14.5 + 25.1i)7-s + 7.99·8-s + 6·10-s + (28.5 − 49.3i)11-s + (−10 − 17.3i)13-s + (−28.9 − 50.2i)14-s + (−8 + 13.8i)16-s − 72·17-s − 106·19-s + (−6.00 + 10.3i)20-s + (57 + 98.7i)22-s + (−87 − 150. i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.134 − 0.232i)5-s + (−0.782 + 1.35i)7-s + 0.353·8-s + 0.189·10-s + (0.781 − 1.35i)11-s + (−0.213 − 0.369i)13-s + (−0.553 − 0.958i)14-s + (−0.125 + 0.216i)16-s − 1.02·17-s − 1.27·19-s + (−0.0670 + 0.116i)20-s + (0.552 + 0.956i)22-s + (−0.788 − 1.36i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(9.55830\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.269700 - 0.321416i\)
\(L(\frac12)\) \(\approx\) \(0.269700 - 0.321416i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (1.5 + 2.59i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (14.5 - 25.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10 + 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 72T + 4.91e3T^{2} \)
19 \( 1 + 106T + 6.85e3T^{2} \)
23 \( 1 + (87 + 150. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-105 + 181. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (23.5 + 40.7i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 2T + 5.06e4T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (109 - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (237 - 410. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 81T + 1.48e5T^{2} \)
59 \( 1 + (42 + 72.7i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (28 - 48.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-71 - 122. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 360T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3T + 3.89e5T^{2} \)
79 \( 1 + (-80 + 138. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (367.5 - 636. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 954T + 7.04e5T^{2} \)
97 \( 1 + (95.5 - 165. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.21663118990706517866106410720, −11.09445837475274361932046946922, −9.849189692860151295927249160823, −8.685214479848840441204267351935, −8.403110357903971472854868387781, −6.39214901652596612966337534116, −6.09165399188575585706751479967, −4.41198108409060805179791286022, −2.59434338006940623333825413966, −0.21477367290383696404551401112, 1.74283343142795829190432850856, 3.59658281919538371279161292043, 4.50902970436037553655730407422, 6.74709521559687043778359784800, 7.24323778176716392059362775706, 8.840950590350059742289396868510, 9.840945962251592108561264170287, 10.53637134011256890900911858651, 11.58786614410119329600216539824, 12.65531501984559689392595195299

Graph of the $Z$-function along the critical line