Properties

Label 2-432-27.14-c2-0-21
Degree $2$
Conductor $432$
Sign $0.998 - 0.0621i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 − 1.50i)3-s + (−0.298 + 0.355i)5-s + (10.1 + 3.69i)7-s + (4.47 − 7.80i)9-s + (10.2 + 12.1i)11-s + (−3.11 + 17.6i)13-s + (−0.240 + 1.37i)15-s + (−22.6 − 13.0i)17-s + (−1.77 − 3.08i)19-s + (31.9 − 5.66i)21-s + (−1.41 − 3.87i)23-s + (4.30 + 24.4i)25-s + (−0.107 − 26.9i)27-s + (41.0 − 7.23i)29-s + (6.62 − 2.41i)31-s + ⋯
L(s)  = 1  + (0.865 − 0.501i)3-s + (−0.0597 + 0.0711i)5-s + (1.44 + 0.527i)7-s + (0.497 − 0.867i)9-s + (0.930 + 1.10i)11-s + (−0.239 + 1.36i)13-s + (−0.0160 + 0.0915i)15-s + (−1.33 − 0.769i)17-s + (−0.0936 − 0.162i)19-s + (1.51 − 0.269i)21-s + (−0.0613 − 0.168i)23-s + (0.172 + 0.976i)25-s + (−0.00397 − 0.999i)27-s + (1.41 − 0.249i)29-s + (0.213 − 0.0777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0621i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.998 - 0.0621i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ 0.998 - 0.0621i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.71247 + 0.0843184i\)
\(L(\frac12)\) \(\approx\) \(2.71247 + 0.0843184i\)
\(L(2)\) 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 \( 1 + (-2.59 + 1.50i)T \)
good5 \( 1 + (0.298 - 0.355i)T + (-4.34 - 24.6i)T^{2} \)
7 \( 1 + (-10.1 - 3.69i)T + (37.5 + 31.4i)T^{2} \)
11 \( 1 + (-10.2 - 12.1i)T + (-21.0 + 119. i)T^{2} \)
13 \( 1 + (3.11 - 17.6i)T + (-158. - 57.8i)T^{2} \)
17 \( 1 + (22.6 + 13.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.77 + 3.08i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (1.41 + 3.87i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (-41.0 + 7.23i)T + (790. - 287. i)T^{2} \)
31 \( 1 + (-6.62 + 2.41i)T + (736. - 617. i)T^{2} \)
37 \( 1 + (-4.92 + 8.53i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (42.8 + 7.56i)T + (1.57e3 + 574. i)T^{2} \)
43 \( 1 + (-27.2 + 22.8i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (-5.51 + 15.1i)T + (-1.69e3 - 1.41e3i)T^{2} \)
53 \( 1 + 75.6iT - 2.80e3T^{2} \)
59 \( 1 + (-18.4 + 21.9i)T + (-604. - 3.42e3i)T^{2} \)
61 \( 1 + (55.9 + 20.3i)T + (2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (-4.03 + 22.8i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (-32.2 - 18.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (26.0 + 45.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-20.9 - 118. i)T + (-5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (115. - 20.2i)T + (6.47e3 - 2.35e3i)T^{2} \)
89 \( 1 + (117. - 67.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (72.3 - 60.6i)T + (1.63e3 - 9.26e3i)T^{2} \)
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

−11.27121556406362348494024955901, −9.733774998492155025967358398790, −8.989959241011711876970791240658, −8.359446220545489214200880071130, −7.14625636928220517238473677198, −6.68647257559057061698059574043, −4.86743405076218568789997573977, −4.15995443043747611913600066587, −2.36267206427955859189443210220, −1.62451175858066846462329549669, 1.27323327467409743166538608276, 2.80789170894966703081720959704, 4.08891756667167677124044056081, 4.81262085955094922096368864048, 6.19056490057549645648855017106, 7.56750892229177944213614519617, 8.427890748674449909953178822735, 8.734054620169568148111022199955, 10.23341521173676477048488589975, 10.74873514370038927880889890059

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