↑

Properties

Label 2-728-13.3-c1-0-17
Degree $2$
Conductor $728$
Sign $0.488 + 0.872i$
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  + (−0.366 − 0.633i)3-s + 3.73·5-s + (−0.5 + 0.866i)7-s + (1.23 − 2.13i)9-s + (−2.36 − 4.09i)11-s + (−0.866 − 3.5i)13-s + (−1.36 − 2.36i)15-s + (0.133 − 0.232i)17-s + (−0.267 + 0.464i)19-s + 0.732·21-s + (2.63 + 4.56i)23-s + 8.92·25-s − 4·27-s + (2.23 + 3.86i)29-s − 0.732·31-s + ⋯
L(s)  = 1  + (−0.211 − 0.366i)3-s + 1.66·5-s + (−0.188 + 0.327i)7-s + (0.410 − 0.711i)9-s + (−0.713 − 1.23i)11-s + (−0.240 − 0.970i)13-s + (−0.352 − 0.610i)15-s + (0.0324 − 0.0562i)17-s + (−0.0614 + 0.106i)19-s + 0.159·21-s + (0.549 + 0.951i)23-s + 1.78·25-s − 0.769·27-s + (0.414 + 0.717i)29-s − 0.131·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.488 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54463 - 0.905051i\)
\(L(\frac12)\) \(\approx\) \(1.54463 - 0.905051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 + 3.5i)T \)
good3 \( 1 + (0.366 + 0.633i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.73T + 5T^{2} \)
11 \( 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.133 + 0.232i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.267 - 0.464i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.63 - 4.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.732T + 31T^{2} \)
37 \( 1 + (-0.232 - 0.401i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.59 + 7.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.36 + 7.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 4.46T + 53T^{2} \)
59 \( 1 + (2.83 - 4.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.59 - 7.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.90 + 3.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.26 - 7.39i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 1.80T + 79T^{2} \)
83 \( 1 - 8.19T + 83T^{2} \)
89 \( 1 + (-4.46 - 7.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.19 - 12.4i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.33376996496200713899322212960, −9.348803804455407979027619769412, −8.784793362120243069998926945784, −7.53246363661199065027094686017, −6.54657487765130124280197397695, −5.65761718687828885706333513991, −5.36356039573484082042674351706, −3.43456142709842879156339738647, −2.41625725786030713833333851414, −1.00784462239675122668398067348, 1.76269430338341625078964154521, 2.58284919452413614959816925097, 4.50481569258165529703646586587, 4.95424097901410503586884103865, 6.12601751937282076365236274045, 6.89344523206410167224027622916, 7.87324067466724270077145679376, 9.243010242832750038581149683999, 9.754294254880652819129800658998, 10.36917485220557118850946264471

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