↑

Properties

Label 2-546-13.3-c1-0-7
Degree $2$
Conductor $546$
Sign $0.859 + 0.511i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 4.27·5-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−2.13 − 3.70i)10-s + (2 + 3.46i)11-s + 0.999·12-s + (3.5 − 0.866i)13-s + 0.999·14-s + (−2.13 − 3.70i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.91·5-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.675 − 1.17i)10-s + (0.603 + 1.04i)11-s + 0.288·12-s + (0.970 − 0.240i)13-s + 0.267·14-s + (−0.551 − 0.955i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45575 - 0.400089i\)
\(L(\frac12)\) \(\approx\) \(1.45575 - 0.400089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 4.27T + 5T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.27 - 5.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.63 - 4.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 + (0.137 + 0.238i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.13 + 7.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.91 + 10.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.54T + 47T^{2} \)
53 \( 1 + 3.54T + 53T^{2} \)
59 \( 1 + (-5.91 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.36 - 2.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.63 - 4.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.72T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (0.362 + 0.627i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.274 + 0.476i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.56035633902108720902277889998, −9.890639418194349868791814596539, −9.196799336838843808314271882911, −8.321354610595611986598243092135, −6.93684760129807240353044800478, −6.09921726721841033672921502160, −5.39608951024021244020050534097, −3.80605076785381613302840167360, −2.12285541535609941235298984318, −1.62184992632624417131358898284, 1.20874050595958855215314506543, 2.90541959188312581433680185504, 4.55626558573150551997766725923, 5.53192785511085364996752855248, 6.39246210159131191168886133129, 6.78993651485837534191489007827, 8.712081039041671103209102551350, 9.023375100545003505405037207302, 9.829748268604782101754423572577, 10.81612686651108537584171496689

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