Properties

Label 2-162-81.5-c2-0-6
Degree $2$
Conductor $162$
Sign $-0.750 - 0.661i$
Analytic cond. $4.41418$
Root an. cond. $2.10099$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.326 + 1.37i)2-s + (2.18 + 2.05i)3-s + (−1.78 + 0.897i)4-s + (−1.03 + 0.772i)5-s + (−2.10 + 3.68i)6-s + (−4.52 + 2.97i)7-s + (−1.81 − 2.16i)8-s + (0.588 + 8.98i)9-s + (−1.40 − 1.17i)10-s + (−0.852 + 7.29i)11-s + (−5.75 − 1.69i)12-s + (0.717 + 2.39i)13-s + (−5.57 − 5.26i)14-s + (−3.85 − 0.436i)15-s + (2.38 − 3.20i)16-s + (12.8 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.163 + 0.688i)2-s + (0.729 + 0.683i)3-s + (−0.446 + 0.224i)4-s + (−0.207 + 0.154i)5-s + (−0.351 + 0.613i)6-s + (−0.647 + 0.425i)7-s + (−0.227 − 0.270i)8-s + (0.0654 + 0.997i)9-s + (−0.140 − 0.117i)10-s + (−0.0774 + 0.662i)11-s + (−0.479 − 0.141i)12-s + (0.0551 + 0.184i)13-s + (−0.398 − 0.375i)14-s + (−0.257 − 0.0290i)15-s + (0.149 − 0.200i)16-s + (0.754 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.750 - 0.661i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.750 - 0.661i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.579342 + 1.53364i\)
\(L(\frac12)\) \(\approx\) \(0.579342 + 1.53364i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.326 - 1.37i)T \)
3 \( 1 + (-2.18 - 2.05i)T \)
good5 \( 1 + (1.03 - 0.772i)T + (7.17 - 23.9i)T^{2} \)
7 \( 1 + (4.52 - 2.97i)T + (19.4 - 44.9i)T^{2} \)
11 \( 1 + (0.852 - 7.29i)T + (-117. - 27.9i)T^{2} \)
13 \( 1 + (-0.717 - 2.39i)T + (-141. + 92.8i)T^{2} \)
17 \( 1 + (-12.8 - 2.26i)T + (271. + 98.8i)T^{2} \)
19 \( 1 + (0.163 + 0.928i)T + (-339. + 123. i)T^{2} \)
23 \( 1 + (-23.9 + 36.3i)T + (-209. - 485. i)T^{2} \)
29 \( 1 + (-10.5 + 9.97i)T + (48.8 - 839. i)T^{2} \)
31 \( 1 + (0.520 - 8.93i)T + (-954. - 111. i)T^{2} \)
37 \( 1 + (-18.2 + 6.63i)T + (1.04e3 - 879. i)T^{2} \)
41 \( 1 + (-2.27 + 9.59i)T + (-1.50e3 - 754. i)T^{2} \)
43 \( 1 + (-2.47 - 5.74i)T + (-1.26e3 + 1.34e3i)T^{2} \)
47 \( 1 + (-64.0 + 3.72i)T + (2.19e3 - 256. i)T^{2} \)
53 \( 1 + (-31.0 - 17.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (12.6 + 108. i)T + (-3.38e3 + 802. i)T^{2} \)
61 \( 1 + (-55.2 - 27.7i)T + (2.22e3 + 2.98e3i)T^{2} \)
67 \( 1 + (-8.81 + 9.34i)T + (-261. - 4.48e3i)T^{2} \)
71 \( 1 + (-9.13 + 10.8i)T + (-875. - 4.96e3i)T^{2} \)
73 \( 1 + (-31.7 + 26.6i)T + (925. - 5.24e3i)T^{2} \)
79 \( 1 + (76.6 - 18.1i)T + (5.57e3 - 2.80e3i)T^{2} \)
83 \( 1 + (0.889 + 3.75i)T + (-6.15e3 + 3.09e3i)T^{2} \)
89 \( 1 + (-7.81 - 9.31i)T + (-1.37e3 + 7.80e3i)T^{2} \)
97 \( 1 + (25.5 - 34.2i)T + (-2.69e3 - 9.01e3i)T^{2} \)
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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

−13.14448311693762407212347997521, −12.30579385795193900956946736143, −10.78658050174289995532716025560, −9.719056128518928116633357227912, −8.929878261083549850761755479780, −7.85727171908937991134329094324, −6.75552621507181357226019924408, −5.32142272317336864980057912150, −4.10354830552362013777558628335, −2.79151174323904941281768777820, 0.965763838215728270979713613647, 2.86779025421993224346103860664, 3.85284000285783488235575952215, 5.70235756250974964381422170061, 7.08778850932232550180586294353, 8.179593776609689806941487350039, 9.232840693063506326557208986957, 10.17173568189999572413222686505, 11.45455770392296862901168594193, 12.36296033579070932868275694628

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