↑

Properties

Label 2-432-4.3-c4-0-19
Degree $2$
Conductor $432$
Sign $0.866 + 0.5i$
Analytic cond. $44.6558$
Root an. cond. $6.68250$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.46·5-s + 54.5i·7-s − 173. i·11-s + 108.·13-s − 228.·17-s − 297. i·19-s + 66.9i·23-s − 618.·25-s + 1.30e3·29-s + 36.6i·31-s + 134. i·35-s + 1.42e3·37-s + 1.15e3·41-s + 1.47e3i·43-s − 642. i·47-s + ⋯
L(s)  = 1  + 0.0987·5-s + 1.11i·7-s − 1.43i·11-s + 0.642·13-s − 0.791·17-s − 0.823i·19-s + 0.126i·23-s − 0.990·25-s + 1.55·29-s + 0.0381i·31-s + 0.109i·35-s + 1.03·37-s + 0.684·41-s + 0.798i·43-s − 0.290i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(44.6558\)
Root analytic conductor: \(6.68250\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :2),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.944313548\)
\(L(\frac12)\) \(\approx\) \(1.944313548\)
\(L(3)\) 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 \( 1 - 2.46T + 625T^{2} \)
7 \( 1 - 54.5iT - 2.40e3T^{2} \)
11 \( 1 + 173. iT - 1.46e4T^{2} \)
13 \( 1 - 108.T + 2.85e4T^{2} \)
17 \( 1 + 228.T + 8.35e4T^{2} \)
19 \( 1 + 297. iT - 1.30e5T^{2} \)
23 \( 1 - 66.9iT - 2.79e5T^{2} \)
29 \( 1 - 1.30e3T + 7.07e5T^{2} \)
31 \( 1 - 36.6iT - 9.23e5T^{2} \)
37 \( 1 - 1.42e3T + 1.87e6T^{2} \)
41 \( 1 - 1.15e3T + 2.82e6T^{2} \)
43 \( 1 - 1.47e3iT - 3.41e6T^{2} \)
47 \( 1 + 642. iT - 4.87e6T^{2} \)
53 \( 1 - 3.66e3T + 7.89e6T^{2} \)
59 \( 1 + 3.35e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.09e3T + 1.38e7T^{2} \)
67 \( 1 + 4.23e3iT - 2.01e7T^{2} \)
71 \( 1 + 9.53e3iT - 2.54e7T^{2} \)
73 \( 1 - 194.T + 2.83e7T^{2} \)
79 \( 1 - 1.00e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.47e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.29e4T + 6.27e7T^{2} \)
97 \( 1 - 1.48e4T + 8.85e7T^{2} \)
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

−10.62099527486515828390649259061, −9.310844095932512484734518740932, −8.735248551777513964211776967259, −7.921176692053755168431247475854, −6.40347284476732418503826365541, −5.89284138285530625407046497808, −4.70523704157013926906182120566, −3.31360151140927092619094870249, −2.28231489149435697506178083207, −0.65935444614057311912253391133, 0.993536554060921550219406624944, 2.28250454029982951335743389298, 3.91631969256360883283877816146, 4.55245458226239358651281353887, 5.98793691983704391799513087169, 6.98331595782187598604721388379, 7.72053142283779245041656785243, 8.803485301237441337813331393061, 9.974064793085964323026171758809, 10.38861550544017154727168859424

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