Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.633 - 0.773i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.78 − 5.36i)2-s + (−25.6 − 19.1i)4-s + (−12.8 + 7.42i)5-s + (−15.2 − 8.83i)7-s + (−148. + 103. i)8-s + (16.9 + 82.2i)10-s + (−143. + 248. i)11-s + (42.2 + 73.1i)13-s + (−74.6 + 66.3i)14-s + (291. + 981. i)16-s + 1.60e3i·17-s − 1.66e3i·19-s + (472. + 55.6i)20-s + (1.07e3 + 1.21e3i)22-s + (2.32e3 + 4.02e3i)23-s + ⋯
L(s)  = 1  + (0.315 − 0.949i)2-s + (−0.801 − 0.598i)4-s + (−0.230 + 0.132i)5-s + (−0.118 − 0.0681i)7-s + (−0.820 + 0.572i)8-s + (0.0535 + 0.260i)10-s + (−0.356 + 0.618i)11-s + (0.0693 + 0.120i)13-s + (−0.101 + 0.0905i)14-s + (0.284 + 0.958i)16-s + 1.35i·17-s − 1.06i·19-s + (0.263 + 0.0311i)20-s + (0.474 + 0.533i)22-s + (0.916 + 1.58i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.633 - 0.773i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.633 - 0.773i)\)
\(L(3)\)  \(\approx\)  \(0.850665 + 0.402831i\)
\(L(\frac12)\)  \(\approx\)  \(0.850665 + 0.402831i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-1.78 + 5.36i)T \)
3 \( 1 \)
good5 \( 1 + (12.8 - 7.42i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (15.2 + 8.83i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (143. - 248. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-42.2 - 73.1i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.60e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.66e3iT - 2.47e6T^{2} \)
23 \( 1 + (-2.32e3 - 4.02e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.41e3 + 1.39e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-2.78e3 + 1.60e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 4.62e3T + 6.93e7T^{2} \)
41 \( 1 + (9.81e3 - 5.66e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.23e4 - 7.15e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.69e3 + 2.93e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 3.10e4iT - 4.18e8T^{2} \)
59 \( 1 + (1.09e4 + 1.88e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.78e4 - 3.09e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.80e4 - 1.04e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.22e4T + 1.80e9T^{2} \)
73 \( 1 + 7.29e4T + 2.07e9T^{2} \)
79 \( 1 + (7.60e4 + 4.39e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.22e4 + 2.11e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 1.19e5iT - 5.58e9T^{2} \)
97 \( 1 + (-3.95e4 + 6.85e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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

−12.97081632619427836372287891537, −11.77382832000712550736110644780, −10.94580732283713102761891893589, −9.893086067762525251788928071025, −8.874055055128859558083373420644, −7.40751324344975791934574802097, −5.78194287574607761162592039895, −4.45380611600197452937841534335, −3.16653045399822261991999344046, −1.57907791984875555763381998413, 0.32807053882336509659253472431, 3.08029739306444890601566988375, 4.57336439660391216009951604978, 5.75667701369187035233069641350, 6.96674275887329566446759605182, 8.117150612154834750640916389080, 9.025190397349879330792401846854, 10.37880433785091906983668681115, 11.86679477048367681912259109759, 12.77213324113386429638692807353

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