Properties

Label 2-162-27.11-c2-0-0
Degree $2$
Conductor $162$
Sign $-0.992 + 0.121i$
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  + (−0.909 + 1.08i)2-s + (−0.347 − 1.96i)4-s + (−2.71 + 7.45i)5-s + (0.0787 − 0.446i)7-s + (2.44 + 1.41i)8-s + (−5.60 − 9.71i)10-s + (−4.82 − 13.2i)11-s + (−9.80 + 8.22i)13-s + (0.412 + 0.491i)14-s + (−3.75 + 1.36i)16-s + (−28.5 + 16.4i)17-s + (0.202 − 0.351i)19-s + (15.6 + 2.75i)20-s + (18.7 + 6.82i)22-s + (−14.2 + 2.51i)23-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.0868 − 0.492i)4-s + (−0.542 + 1.49i)5-s + (0.0112 − 0.0638i)7-s + (0.306 + 0.176i)8-s + (−0.560 − 0.971i)10-s + (−0.438 − 1.20i)11-s + (−0.754 + 0.632i)13-s + (0.0294 + 0.0351i)14-s + (−0.234 + 0.0855i)16-s + (−1.67 + 0.969i)17-s + (0.0106 − 0.0184i)19-s + (0.781 + 0.137i)20-s + (0.851 + 0.310i)22-s + (−0.620 + 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.992 + 0.121i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.992 + 0.121i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0280720 - 0.459334i\)
\(L(\frac12)\) \(\approx\) \(0.0280720 - 0.459334i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 1.08i)T \)
3 \( 1 \)
good5 \( 1 + (2.71 - 7.45i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (-0.0787 + 0.446i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (4.82 + 13.2i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (9.80 - 8.22i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (28.5 - 16.4i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.202 + 0.351i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (14.2 - 2.51i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (-16.8 + 20.0i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (-4.33 - 24.6i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (-3.84 - 6.65i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-15.9 - 18.9i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (16.8 - 6.13i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (-46.7 - 8.24i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 + 0.261iT - 2.80e3T^{2} \)
59 \( 1 + (18.8 - 51.7i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (18.1 - 103. i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (49.4 - 41.5i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (-94.6 + 54.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-31.4 + 54.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (14.7 + 12.3i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (36.7 - 43.7i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (89.7 + 51.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (52.8 - 19.2i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−13.48631611510887017072000931089, −11.85148514421102832068251767980, −10.91368542652268022826548016099, −10.33333948452022505337115288125, −8.891990800115230954460721474696, −7.86738555516189984379530783013, −6.85333048764887073199677694459, −6.05353343661406897109692758251, −4.20181889055919782499753934940, −2.61165552198302530288611218134, 0.31465124989319166502243345236, 2.26265412362668713976221529728, 4.31188691491707283243041749689, 5.09033405697291459696140213655, 7.15861099410545320153132591707, 8.150039904043956704380020969829, 9.100322415200566783694846638916, 9.926145281017607164666534436254, 11.20300776602160729408784649401, 12.32826241557504166561187451655

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