Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $0.829 - 0.557i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 + 3.82i)2-s + (−13.2 + 8.92i)4-s + 39.0·5-s − 12.2i·7-s + (−49.6 − 40.4i)8-s + (45.5 + 149. i)10-s − 111. i·11-s + 208.·13-s + (46.6 − 14.2i)14-s + (96.6 − 237. i)16-s − 93.3·17-s − 26.8i·19-s + (−518. + 348. i)20-s + (424. − 129. i)22-s − 874. i·23-s + ⋯
L(s)  = 1  + (0.291 + 0.956i)2-s + (−0.829 + 0.557i)4-s + 1.56·5-s − 0.249i·7-s + (−0.775 − 0.631i)8-s + (0.455 + 1.49i)10-s − 0.917i·11-s + 1.23·13-s + (0.238 − 0.0726i)14-s + (0.377 − 0.925i)16-s − 0.323·17-s − 0.0744i·19-s + (−1.29 + 0.871i)20-s + (0.877 − 0.267i)22-s − 1.65i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $0.829 - 0.557i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ 0.829 - 0.557i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.899328317\)
\(L(\frac12)\)  \(\approx\)  \(2.899328317\)
\(L(3)\)   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 \( 1 + (-1.16 - 3.82i)T \)
3 \( 1 \)
good5 \( 1 - 39.0T + 625T^{2} \)
7 \( 1 + 12.2iT - 2.40e3T^{2} \)
11 \( 1 + 111. iT - 1.46e4T^{2} \)
13 \( 1 - 208.T + 2.85e4T^{2} \)
17 \( 1 + 93.3T + 8.35e4T^{2} \)
19 \( 1 + 26.8iT - 1.30e5T^{2} \)
23 \( 1 + 874. iT - 2.79e5T^{2} \)
29 \( 1 - 1.30e3T + 7.07e5T^{2} \)
31 \( 1 + 685. iT - 9.23e5T^{2} \)
37 \( 1 + 1.76e3T + 1.87e6T^{2} \)
41 \( 1 + 78.0T + 2.82e6T^{2} \)
43 \( 1 + 1.62e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.30e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.31e3T + 7.89e6T^{2} \)
59 \( 1 - 5.56e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.18e3T + 1.38e7T^{2} \)
67 \( 1 - 246. iT - 2.01e7T^{2} \)
71 \( 1 - 4.60e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.56e3T + 2.83e7T^{2} \)
79 \( 1 - 5.18e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.87e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.16e3T + 6.27e7T^{2} \)
97 \( 1 + 5.73e3T + 8.85e7T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.83548381800524200819765191604, −10.05132077691533264459492855345, −8.848695874295487726568668489933, −8.447105533523974079823091956778, −6.82946447804606150925652136994, −6.17186253018194560190373077652, −5.43440008585809505924371184486, −4.14084911315653573750851738665, −2.69993019005585562029312801632, −0.897186080992715486453342949868, 1.32383160913836763596528908516, 2.12806342895558660268055648110, 3.43922923471703314652389210188, 4.89624156797015776154541058677, 5.73025108533868679362425105061, 6.69021600770837742602337021187, 8.490406914101075668369614426620, 9.324384142233706887207037362888, 10.04219584684015950645292862286, 10.76081960061673723242046836105

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