↑

Properties

Label 2-546-39.32-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.822 - 0.568i$
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  + (−0.965 + 0.258i)2-s + (0.458 + 1.67i)3-s + (0.866 − 0.499i)4-s + (1.24 + 1.24i)5-s + (−0.875 − 1.49i)6-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (−2.57 + 1.53i)9-s + (−1.52 − 0.879i)10-s + (0.417 + 1.55i)11-s + (1.23 + 1.21i)12-s + (−3.43 − 1.10i)13-s − i·14-s + (−1.50 + 2.64i)15-s + (0.500 − 0.866i)16-s + (2.60 + 4.50i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.264 + 0.964i)3-s + (0.433 − 0.249i)4-s + (0.556 + 0.556i)5-s + (−0.357 − 0.610i)6-s + (−0.0978 + 0.365i)7-s + (−0.249 + 0.249i)8-s + (−0.859 + 0.510i)9-s + (−0.481 − 0.278i)10-s + (0.125 + 0.470i)11-s + (0.355 + 0.351i)12-s + (−0.952 − 0.305i)13-s − 0.267i·14-s + (−0.389 + 0.683i)15-s + (0.125 − 0.216i)16-s + (0.631 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305757 + 0.979791i\)
\(L(\frac12)\) \(\approx\) \(0.305757 + 0.979791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.458 - 1.67i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (3.43 + 1.10i)T \)
good5 \( 1 + (-1.24 - 1.24i)T + 5iT^{2} \)
11 \( 1 + (-0.417 - 1.55i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.60 - 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.82 + 0.490i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.39 - 2.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.73 - 3.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.89 + 1.89i)T - 31iT^{2} \)
37 \( 1 + (0.216 - 0.0580i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.320 + 0.0859i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.90 - 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.55 - 5.55i)T - 47iT^{2} \)
53 \( 1 + 6.37iT - 53T^{2} \)
59 \( 1 + (5.66 + 1.51i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.51 - 6.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.05 - 7.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.213 - 0.796i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.64 + 4.64i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + (0.280 + 0.280i)T + 83iT^{2} \)
89 \( 1 + (-2.21 - 8.25i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.03 + 0.276i)T + (84.0 + 48.5i)T^{2} \)
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

−10.71946100147237459000190552406, −10.06392892232016417147413130142, −9.659155316374163163179679048641, −8.590349810259979288129470102041, −7.82967648921986870191172252040, −6.59447792677170333389391090630, −5.71780815014170394364196278775, −4.63145090327996537477977491210, −3.18233494812093514342844282121, −2.12866671295953823354161603541, 0.69845851038728081154328450404, 2.02459732698967890207973993494, 3.16808378117415369344670083502, 4.89032567010811983074590495616, 6.12028961003615152248201958329, 6.98542012972624520757085430163, 7.83400978059981270019184750148, 8.680568574375175511818134151003, 9.478212280821465054984768444046, 10.23133212478053124674460498448

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