↑

Properties

Label 2-723-723.194-c0-0-0
Degree $2$
Conductor $723$
Sign $0.0579 - 0.998i$
Analytic cond. $0.360824$
Root an. cond. $0.600686$
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  + (0.156 + 0.987i)3-s + i·4-s + (1.04 − 0.0819i)7-s + (−0.951 + 0.309i)9-s + (−0.987 + 0.156i)12-s + (1.65 − 1.01i)13-s − 16-s + (−1.57 + 0.652i)19-s + (0.243 + 1.01i)21-s + (−0.951 − 0.309i)25-s + (−0.453 − 0.891i)27-s + (0.0819 + 1.04i)28-s + (−0.581 − 0.497i)31-s + (−0.309 − 0.951i)36-s + (0.133 + 0.0819i)37-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)3-s + i·4-s + (1.04 − 0.0819i)7-s + (−0.951 + 0.309i)9-s + (−0.987 + 0.156i)12-s + (1.65 − 1.01i)13-s − 16-s + (−1.57 + 0.652i)19-s + (0.243 + 1.01i)21-s + (−0.951 − 0.309i)25-s + (−0.453 − 0.891i)27-s + (0.0819 + 1.04i)28-s + (−0.581 − 0.497i)31-s + (−0.309 − 0.951i)36-s + (0.133 + 0.0819i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0579 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0579 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.0579 - 0.998i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ 0.0579 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071896082\)
\(L(\frac12)\) \(\approx\) \(1.071896082\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.156 - 0.987i)T \)
241 \( 1 + (-0.156 - 0.987i)T \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (-1.04 + 0.0819i)T + (0.987 - 0.156i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-1.65 + 1.01i)T + (0.453 - 0.891i)T^{2} \)
17 \( 1 + (-0.156 - 0.987i)T^{2} \)
19 \( 1 + (1.57 - 0.652i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.891 + 0.453i)T^{2} \)
29 \( 1 + (-0.587 - 0.809i)T^{2} \)
31 \( 1 + (0.581 + 0.497i)T + (0.156 + 0.987i)T^{2} \)
37 \( 1 + (-0.133 - 0.0819i)T + (0.453 + 0.891i)T^{2} \)
41 \( 1 + (-0.587 + 0.809i)T^{2} \)
43 \( 1 + (-0.101 + 1.29i)T + (-0.987 - 0.156i)T^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 - 0.309i)T^{2} \)
61 \( 1 + (-1.87 + 0.297i)T + (0.951 - 0.309i)T^{2} \)
67 \( 1 + (1.04 - 0.533i)T + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.987 - 0.156i)T^{2} \)
73 \( 1 + (-1.29 - 0.794i)T + (0.453 + 0.891i)T^{2} \)
79 \( 1 + (-1.59 - 0.253i)T + (0.951 + 0.309i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
97 \( 1 + (-0.297 + 0.0966i)T + (0.809 - 0.587i)T^{2} \)
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

−10.94305718226880043375051724836, −10.08562071937956301862459581525, −8.805329021233705426722239030875, −8.340521363931747476406149655961, −7.78407173491919240032760067821, −6.27959828772781708107548095956, −5.30705440964049418337020722026, −4.09051853809155315265060817140, −3.66638518007635575430241790015, −2.22828770461585235388507943151, 1.39667898996689312948828851651, 2.15812642129919159558977290088, 3.98835235732178421478110486989, 5.13012263448234144710584106099, 6.20055643019387521766243993655, 6.65151127370262490299009817224, 7.892222996989722216709201369650, 8.690742889731433257046317931542, 9.294982452861592163252909324353, 10.76691329769042299729970889731

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