↑

Properties

Label 2-728-104.101-c1-0-20
Degree $2$
Conductor $728$
Sign $0.836 - 0.548i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
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  + (0.526 − 1.31i)2-s + (−1.82 + 1.05i)3-s + (−1.44 − 1.38i)4-s + 1.99·5-s + (0.421 + 2.94i)6-s + (0.866 + 0.5i)7-s + (−2.57 + 1.16i)8-s + (0.714 − 1.23i)9-s + (1.04 − 2.61i)10-s + (1.16 + 2.01i)11-s + (4.08 + 0.998i)12-s + (−3.55 + 0.599i)13-s + (1.11 − 0.873i)14-s + (−3.62 + 2.09i)15-s + (0.177 + 3.99i)16-s + (0.978 − 1.69i)17-s + ⋯
L(s)  = 1  + (0.372 − 0.928i)2-s + (−1.05 + 0.607i)3-s + (−0.722 − 0.691i)4-s + 0.890·5-s + (0.171 + 1.20i)6-s + (0.327 + 0.188i)7-s + (−0.910 + 0.413i)8-s + (0.238 − 0.412i)9-s + (0.331 − 0.826i)10-s + (0.351 + 0.608i)11-s + (1.18 + 0.288i)12-s + (−0.986 + 0.166i)13-s + (0.297 − 0.233i)14-s + (−0.937 + 0.541i)15-s + (0.0442 + 0.999i)16-s + (0.237 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.836 - 0.548i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08407 + 0.323644i\)
\(L(\frac12)\) \(\approx\) \(1.08407 + 0.323644i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.526 + 1.31i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.55 - 0.599i)T \)
good3 \( 1 + (1.82 - 1.05i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.99T + 5T^{2} \)
11 \( 1 + (-1.16 - 2.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.978 + 1.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.12 - 5.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.75 - 4.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.43 + 3.71i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.25iT - 31T^{2} \)
37 \( 1 + (-5.03 - 8.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.36 - 2.51i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.68 - 5.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.89iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 + (1.32 - 2.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.08 - 5.24i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.222 + 0.385i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.82 + 2.78i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.74iT - 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 0.107T + 83T^{2} \)
89 \( 1 + (-5.46 + 3.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.896 + 0.517i)T + (48.5 + 84.0i)T^{2} \)
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

−10.32660033914202987584781040819, −9.975534878916930399922898579414, −9.331476763215344991807939192063, −8.036604622561022185730255315677, −6.49718237764508799628677540358, −5.71485478073911302036781774166, −4.94403984312485647982656998006, −4.28734548890010981453568435719, −2.71031761802793769840549414108, −1.50582194257332289191586425645, 0.61888464075597099039019800497, 2.55473523924125258287378537376, 4.25256979229593105563538563748, 5.26761426829822087960616011285, 5.89033263334008630061845814533, 6.69376817965036988235727865571, 7.28628140356631223548308551872, 8.512703812845808305093785910901, 9.255812816074319806997196316286, 10.39674488194613260662592196617

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