↑

Properties

Label 2-572-13.3-c1-0-3
Degree $2$
Conductor $572$
Sign $0.872 - 0.488i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + (−0.119 − 0.207i)3-s + 3.94·5-s + (−2.21 + 3.82i)7-s + (1.47 − 2.54i)9-s + (0.5 + 0.866i)11-s + (2.5 + 2.59i)13-s + (−0.471 − 0.816i)15-s + (−2.06 + 3.57i)17-s + (−0.619 + 1.07i)19-s + 1.05·21-s + (−2.82 − 4.89i)23-s + 10.5·25-s − 1.42·27-s + (−1.94 − 3.37i)29-s + 1.95·31-s + ⋯
L(s)  = 1  + (−0.0690 − 0.119i)3-s + 1.76·5-s + (−0.835 + 1.44i)7-s + (0.490 − 0.849i)9-s + (0.150 + 0.261i)11-s + (0.693 + 0.720i)13-s + (−0.121 − 0.210i)15-s + (−0.500 + 0.866i)17-s + (−0.142 + 0.246i)19-s + 0.230·21-s + (−0.588 − 1.01i)23-s + 2.10·25-s − 0.273·27-s + (−0.362 − 0.627i)29-s + 0.351·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77089 + 0.462361i\)
\(L(\frac12)\) \(\approx\) \(1.77089 + 0.462361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good3 \( 1 + (0.119 + 0.207i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.94T + 5T^{2} \)
7 \( 1 + (2.21 - 3.82i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (2.06 - 3.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.619 - 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.94 + 3.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.95T + 31T^{2} \)
37 \( 1 + (-5.65 - 9.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.06 + 3.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.83 + 10.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.66T + 47T^{2} \)
53 \( 1 - 0.535T + 53T^{2} \)
59 \( 1 + (4.94 - 8.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.612 + 1.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.01 - 5.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.29 + 10.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 2.46T + 83T^{2} \)
89 \( 1 + (8.91 + 15.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.978 - 1.69i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.52966326687672251277360144675, −9.786495241663671585684404674399, −9.151473524100935279533591920283, −8.580041834325745999013919856298, −6.68478098304469566515728589765, −6.22840862925607113671312511113, −5.68588568596341240854600143896, −4.16119764073536957347489721034, −2.62995502362101678239244085635, −1.72041292009131241352783617731, 1.20806776865288831794180790294, 2.65022169667188793686340609877, 3.98739227565317839536299742195, 5.21138192646605211975012087003, 6.10709243952533816519928243722, 6.93780192837401790640417315526, 7.87469269756496169918237265520, 9.386957078376093334529368730686, 9.698600378410792049663867020472, 10.67361665603951820052779463572

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