Properties

Label 2-726-11.5-c1-0-9
Degree $2$
Conductor $726$
Sign $0.882 - 0.469i$
Analytic cond. $5.79713$
Root an. cond. $2.40772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (1.61 − 1.17i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·12-s + (1.23 + 3.80i)13-s + (1.61 + 1.17i)14-s + (0.309 − 0.951i)16-s + (1.85 − 5.70i)17-s + (−0.809 + 0.587i)18-s + (−3.23 − 2.35i)19-s − 2·21-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.126 − 0.388i)6-s + (0.611 − 0.444i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + 0.288·12-s + (0.342 + 1.05i)13-s + (0.432 + 0.314i)14-s + (0.0772 − 0.237i)16-s + (0.449 − 1.38i)17-s + (−0.190 + 0.138i)18-s + (−0.742 − 0.539i)19-s − 0.436·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.882 - 0.469i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.882 - 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48012 + 0.369476i\)
\(L(\frac12)\) \(\approx\) \(1.48012 + 0.369476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good5 \( 1 + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.23 - 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.85 + 5.70i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.47 - 7.60i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.09 + 5.87i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.85 - 3.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-4.85 - 3.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.47 - 7.60i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-1.85 + 5.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.61 + 1.17i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.32 + 13.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.70 + 11.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-4.32 - 13.3i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.65331282717296094468185497636, −9.381673342435122728088306075413, −8.691659040395518037342758788339, −7.57098931640386838551832689355, −6.97686897619055096831659596385, −6.18345193521894446218958230941, −4.92893368597069613720131500040, −4.48075935633408025933639025977, −2.87015722187842027400982331087, −1.08184591226553115840982858432, 1.13643197792719550989874134069, 2.65106637520017102663001367327, 3.83518993371636689364007079746, 4.83519058430739381297876480849, 5.67652766325058390640449832940, 6.51595497315274371819400239303, 8.106281515319968784669713522478, 8.555134410757491799142193076658, 9.776817584972162389607369622925, 10.54840296141053349207030373624

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