↑

Properties

Label 2-726-11.9-c1-0-2
Degree $2$
Conductor $726$
Sign $0.0219 - 0.999i$
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 + (1.11 + 3.44i)5-s + (0.309 + 0.951i)6-s + (0.5 + 0.363i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 3.61·10-s + 0.999·12-s + (−1 + 3.07i)13-s + (0.5 − 0.363i)14-s + (−2.92 − 2.12i)15-s + (0.309 + 0.951i)16-s + (−1.23 − 3.80i)17-s + (−0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.499 + 1.53i)5-s + (0.126 + 0.388i)6-s + (0.188 + 0.137i)7-s + (−0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + 1.14·10-s + 0.288·12-s + (−0.277 + 0.853i)13-s + (0.133 − 0.0970i)14-s + (−0.755 − 0.549i)15-s + (0.0772 + 0.237i)16-s + (−0.299 − 0.922i)17-s + (−0.190 − 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811470 + 0.793822i\)
\(L(\frac12)\) \(\approx\) \(0.811470 + 0.793822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 3.44i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.5 - 0.363i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1 - 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.23 + 3.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.61 - 3.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + (-5.54 - 4.02i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.04 - 3.21i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.23 - 5.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.61 - 3.35i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.76T + 43T^{2} \)
47 \( 1 + (3.23 - 2.35i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.427 + 1.31i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.14 - 3.52i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + (-1.85 - 5.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.7 - 8.55i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.11 + 6.51i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.5 + 1.53i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (2.97 - 9.14i)T + (-78.4 - 57.0i)T^{2} \)
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

−10.57288458107427234821303505963, −10.06435078906229313108697102984, −9.303051178318629343210944937503, −8.063020483300624260193717349643, −6.71422133584690362108552903982, −6.32013866697431533317911527759, −5.09295329796660562423486390326, −4.06815119901038188035454398914, −2.93059682466465899266495620449, −1.94619264070086552630308397106, 0.56415874457762768693503691625, 2.12153445585720855073901113754, 4.16883658585160758450694764115, 4.83478918524199492108805496043, 5.80666530529369096544080264836, 6.35102526947392918778224361949, 7.75561712077410985911936503003, 8.286829426633319975410202213778, 9.144339542415935020743661242158, 10.06986491708908919558463351562

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