↑

Properties

Label 2-546-13.10-c1-0-3
Degree $2$
Conductor $546$
Sign $0.667 - 0.744i$
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 − 0.332i·5-s + (−0.866 − 0.499i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.166 + 0.288i)10-s + (2.26 − 1.30i)11-s + 0.999·12-s + (3.41 − 1.16i)13-s + 0.999·14-s + (0.288 − 0.166i)15-s + (−0.5 − 0.866i)16-s + (−1.94 + 3.36i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.148i·5-s + (−0.353 − 0.204i)6-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0526 + 0.0911i)10-s + (0.681 − 0.393i)11-s + 0.288·12-s + (0.946 − 0.323i)13-s + 0.267·14-s + (0.0744 − 0.0429i)15-s + (−0.125 − 0.216i)16-s + (−0.471 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13165 + 0.505660i\)
\(L(\frac12)\) \(\approx\) \(1.13165 + 0.505660i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.41 + 1.16i)T \)
good5 \( 1 + 0.332iT - 5T^{2} \)
11 \( 1 + (-2.26 + 1.30i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.94 - 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.85 - 2.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.10 - 3.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.593 - 1.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.07iT - 31T^{2} \)
37 \( 1 + (-0.499 + 0.288i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.451 + 0.260i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.53 + 2.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 + (-9.07 - 5.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.71 - 6.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 5.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.818 + 0.472i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.66iT - 73T^{2} \)
79 \( 1 + 0.943T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 + (-2.92 + 1.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.03 + 2.90i)T + (48.5 + 84.0i)T^{2} \)
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

−10.83355609530452157101228701212, −9.839699141875592161504024291142, −9.144509710331623613908121675418, −8.400953944253608679913008887350, −7.50268761316838694964059406810, −6.34658706230233626192044419964, −5.56977303727984322204221486427, −4.15570252696066319005374244641, −3.14060414376862508795376117042, −1.27720965018775831276361634425, 1.09057123630961569093770172167, 2.54695127516940982732816701043, 3.57077956221773455200562312936, 5.02033403673555551678467121960, 6.67223679115452552462770118025, 6.88884640513958376694074075719, 8.207444072067814404740625425372, 9.015570408264289140388369077208, 9.567062965184724702666482549688, 10.75164477818556914080396249984

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