↑

Properties

Label 2-546-91.25-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.00340 + 0.999i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (2.18 − 1.26i)5-s − 0.999i·6-s + (−1.47 − 2.19i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.26 − 2.18i)10-s + (4.88 + 2.82i)11-s + (−0.499 − 0.866i)12-s + (−3.13 + 1.78i)13-s + (−2.37 − 1.16i)14-s − 2.52i·15-s + (−0.5 − 0.866i)16-s + (0.123 − 0.214i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (0.976 − 0.563i)5-s − 0.408i·6-s + (−0.556 − 0.830i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.398 − 0.690i)10-s + (1.47 + 0.850i)11-s + (−0.144 − 0.249i)12-s + (−0.869 + 0.493i)13-s + (−0.634 − 0.311i)14-s − 0.650i·15-s + (−0.125 − 0.216i)16-s + (0.0300 − 0.0519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00340 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00340 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.00340 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.00340 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74890 - 1.75485i\)
\(L(\frac12)\) \(\approx\) \(1.74890 - 1.75485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.47 + 2.19i)T \)
13 \( 1 + (3.13 - 1.78i)T \)
good5 \( 1 + (-2.18 + 1.26i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.88 - 2.82i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.123 + 0.214i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.70 - 1.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.80 + 3.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + (-5.30 - 3.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.01 + 4.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.59iT - 41T^{2} \)
43 \( 1 - 5.56T + 43T^{2} \)
47 \( 1 + (-5.78 + 3.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.28 - 5.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.89 + 3.97i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.39 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.5 - 6.09i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.06iT - 71T^{2} \)
73 \( 1 + (6.81 + 3.93i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.22 - 5.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.662iT - 83T^{2} \)
89 \( 1 + (4.86 - 2.81i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.8iT - 97T^{2} \)
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

−10.51463986244091829742242876983, −9.523749801424637218379790527894, −9.292328647427683371462303066606, −7.71451274434396492128328034774, −6.67403787533775443359364412697, −6.16259155833391021639882890649, −4.71196049769353842008465122454, −3.92280660484104105408865221362, −2.36557320225474341248318481163, −1.31977872919570090902761125662, 2.29359339384885112555973728525, 3.20364629258551445349692156894, 4.36073782053483567572511849246, 5.82556099407861533955238706017, 6.05490136742601297177733154946, 7.20087504938312730242152138616, 8.503620500053025437944908098945, 9.345339518427348518734174360789, 9.948693717772826610000946051085, 11.07972189883646827075881000447

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