↑

Properties

Label 2-728-7.4-c1-0-6
Degree $2$
Conductor $728$
Sign $0.386 - 0.922i$
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  + (0.5 − 0.866i)3-s + (1.5 + 2.59i)5-s + (−2 + 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s − 13-s + 3·15-s + (−3.5 + 6.06i)17-s + (0.5 + 0.866i)19-s + (0.499 + 2.59i)21-s + (−0.5 − 0.866i)23-s + (−2 + 3.46i)25-s + 5·27-s − 2·29-s + (−4.5 + 7.79i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.670 + 1.16i)5-s + (−0.755 + 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s − 0.277·13-s + 0.774·15-s + (−0.848 + 1.47i)17-s + (0.114 + 0.198i)19-s + (0.109 + 0.566i)21-s + (−0.104 − 0.180i)23-s + (−0.400 + 0.692i)25-s + 0.962·27-s − 0.371·29-s + (−0.808 + 1.39i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37220 + 0.912767i\)
\(L(\frac12)\) \(\approx\) \(1.37220 + 0.912767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
13 \( 1 + T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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

−10.63093029142436353287112069061, −9.731865444256721017122381481268, −8.840313978190087346445080687555, −7.969090184181778678623087228087, −6.78172719457631538530772791062, −6.40277559528180129545465932433, −5.44419339073611806752710635074, −3.81015905313302725879341380622, −2.74511818347429415360205142110, −1.87788654386522647609214175329, 0.837591603142460961702535837250, 2.45128867925479836039053771455, 3.96437978248850991825437624709, 4.56076225428856058876833923460, 5.66526736330495188913803836571, 6.79033916091804252130232647924, 7.52924624440617576593640281936, 9.021570395243144228664604523765, 9.465413251542432290872909183943, 9.737235304556369366034061668449

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