↑

Properties

Label 2-3456-72.29-c0-0-1
Degree $2$
Conductor $3456$
Sign $0.766 + 0.642i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)11-s − 1.73i·17-s + 1.73i·19-s + (0.5 − 0.866i)25-s + (1.5 − 0.866i)41-s + (−1.5 − 0.866i)43-s + (0.5 + 0.866i)49-s + (−0.5 − 0.866i)59-s + (1.5 − 0.866i)67-s + 73-s + (−1 + 1.73i)83-s + (0.5 − 0.866i)97-s + 107-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)11-s − 1.73i·17-s + 1.73i·19-s + (0.5 − 0.866i)25-s + (1.5 − 0.866i)41-s + (−1.5 − 0.866i)43-s + (0.5 + 0.866i)49-s + (−0.5 − 0.866i)59-s + (1.5 − 0.866i)67-s + 73-s + (−1 + 1.73i)83-s + (0.5 − 0.866i)97-s + 107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279485121\)
\(L(\frac12)\) \(\approx\) \(1.279485121\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.644213469869758570584450855969, −8.012819231601090095670454982113, −7.20826179708431050066878284245, −6.42882437727807195683798929457, −5.70858645527314271573276314159, −4.93516927243995895086733339835, −3.94520322877779860275487980809, −3.20518936043917020134117598956, −2.18250585481861927429406102290, −0.854048411930490116078026361986, 1.30407242906896992380656306589, 2.32506809525869714113412386919, 3.38214824166295899304454607174, 4.30108289089354573342380617247, 4.93509740784691288423472396360, 5.93489005047980114802776955175, 6.69939018977804379355384846298, 7.26296576700805143653596249239, 8.175273085706973576322406737780, 8.866730220431212700517573639702

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