Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{3} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·5-s − 3.87i·7-s + 1.73·11-s − 2·13-s + 4.47i·17-s + 6.92·23-s − 4.47i·29-s + 3.87i·31-s + 8.66·35-s + 4·37-s − 8.94i·41-s − 7.74i·43-s + 3.46·47-s − 8.00·49-s + 2.23i·53-s + ⋯
L(s)  = 1  + 0.999i·5-s − 1.46i·7-s + 0.522·11-s − 0.554·13-s + 1.08i·17-s + 1.44·23-s − 0.830i·29-s + 0.695i·31-s + 1.46·35-s + 0.657·37-s − 1.39i·41-s − 1.18i·43-s + 0.505·47-s − 1.14·49-s + 0.307i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1728\)    =    \(2^{6} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1728} (1727, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1728,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.757916897\)
\(L(\frac12)\)  \(\approx\)  \(1.757916897\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 \)
good5 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 + 3.87iT - 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 3.87iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 7.74iT - 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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

−9.432189167915631793471249139414, −8.485911910800638601791973140238, −7.46954256416573118190085433158, −7.00220041392202997163361613404, −6.38919823661287754775416674235, −5.20441920637936385286345674552, −4.09474941180624161284134508973, −3.52744450935604959873829794120, −2.34910766146292448620771132608, −0.912784669434568233243060684461, 0.972589239802239879566466175296, 2.30953153934431804660223957894, 3.20812438538059804743906551918, 4.74006368677779757328994762920, 5.03706830281016540122094677830, 5.99545367709765374992708686670, 6.89300381961172343876538474741, 7.901930991420573274841940002160, 8.715245056520014583456357927119, 9.295907128106874251754033068989

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