Properties

Degree 2
Conductor $ 2 \cdot 3^{4} $
Sign $-0.939 + 0.342i$
Motivic weight 3
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 + (−3 − 5.19i)5-s + (8 − 13.8i)7-s − 7.99·8-s − 12·10-s + (−6 + 10.3i)11-s + (−19 − 32.9i)13-s + (−15.9 − 27.7i)14-s + (−8 + 13.8i)16-s − 126·17-s + 20·19-s + (−12.0 + 20.7i)20-s + (12 + 20.7i)22-s + (−84 − 145. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.268 − 0.464i)5-s + (0.431 − 0.748i)7-s − 0.353·8-s − 0.379·10-s + (−0.164 + 0.284i)11-s + (−0.405 − 0.702i)13-s + (−0.305 − 0.529i)14-s + (−0.125 + 0.216i)16-s − 1.79·17-s + 0.241·19-s + (−0.134 + 0.232i)20-s + (0.116 + 0.201i)22-s + (−0.761 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(162\)    =    \(2 \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.939 + 0.342i$
motivic weight  =  \(3\)
character  :  $\chi_{162} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 162,\ (\ :3/2),\ -0.939 + 0.342i)\)
\(L(2)\)  \(\approx\)  \(0.236027 - 1.33857i\)
\(L(\frac12)\)  \(\approx\)  \(0.236027 - 1.33857i\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-8 + 13.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (6 - 10.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (19 + 32.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 126T + 4.91e3T^{2} \)
19 \( 1 - 20T + 6.85e3T^{2} \)
23 \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (15 - 25.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-44 - 76.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 254T + 5.06e4T^{2} \)
41 \( 1 + (21 + 36.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-26 + 45.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-48 + 83.1i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 198T + 1.48e5T^{2} \)
59 \( 1 + (-330 - 571. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-269 + 465. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (442 + 765. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 792T + 3.57e5T^{2} \)
73 \( 1 - 218T + 3.89e5T^{2} \)
79 \( 1 + (-260 + 450. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-246 + 426. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 810T + 7.04e5T^{2} \)
97 \( 1 + (577 - 999. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.05375373847283134425463339839, −10.91977891829827884120185507012, −10.24899224800968626929539845872, −8.923259017835169994119256918490, −7.87840553880654713249005694560, −6.53150459804077004334445757144, −4.91159470364095355892009719459, −4.15641517158807286216801301998, −2.38185036243216017988320611905, −0.55414160022227999658429454210, 2.38071108501237782494702519733, 4.03230660664965857454205448834, 5.29398596060414535235289714627, 6.47152271349912096464703352360, 7.51158424068889554486111825326, 8.611895914387897786114526314319, 9.582233457319540021251574062611, 11.21399557918409339836763926114, 11.71859973283057404540896298569, 13.03671759598096539007792648426

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