↑

Properties

Label 2-546-273.185-c1-0-29
Degree $2$
Conductor $546$
Sign $-0.963 + 0.267i$
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 + (−0.0864 − 1.72i)3-s + (0.499 + 0.866i)4-s + (−0.886 − 1.53i)5-s + (−0.790 + 1.54i)6-s + (2.61 + 0.424i)7-s − 0.999i·8-s + (−2.98 + 0.299i)9-s + 1.77i·10-s − 0.893i·11-s + (1.45 − 0.939i)12-s + (1.84 − 3.09i)13-s + (−2.04 − 1.67i)14-s + (−2.58 + 1.66i)15-s + (−0.5 + 0.866i)16-s + (−3.74 − 6.49i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.0499 − 0.998i)3-s + (0.249 + 0.433i)4-s + (−0.396 − 0.686i)5-s + (−0.322 + 0.629i)6-s + (0.987 + 0.160i)7-s − 0.353i·8-s + (−0.995 + 0.0997i)9-s + 0.560i·10-s − 0.269i·11-s + (0.419 − 0.271i)12-s + (0.511 − 0.859i)13-s + (−0.547 − 0.447i)14-s + (−0.666 + 0.430i)15-s + (−0.125 + 0.216i)16-s + (−0.909 − 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.963 + 0.267i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.963 + 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116351 - 0.853896i\)
\(L(\frac12)\) \(\approx\) \(0.116351 - 0.853896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0864 + 1.72i)T \)
7 \( 1 + (-2.61 - 0.424i)T \)
13 \( 1 + (-1.84 + 3.09i)T \)
good5 \( 1 + (0.886 + 1.53i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.893iT - 11T^{2} \)
17 \( 1 + (3.74 + 6.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.805iT - 19T^{2} \)
23 \( 1 + (-4.72 - 2.72i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.62 - 4.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.20 + 4.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.84 - 3.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.47 - 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.93 + 3.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.19 + 7.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.72 - 1.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.67 + 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.21iT - 61T^{2} \)
67 \( 1 + 7.98T + 67T^{2} \)
71 \( 1 + (-1.97 - 1.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.54 + 4.35i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.40 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-7.75 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.24 + 2.45i)T + (48.5 + 84.0i)T^{2} \)
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

−10.69257288953141607533260939962, −9.058445360909426794706415983458, −8.751395508825170943285279309327, −7.69698276723467080937810793422, −7.23245683931100357389060594395, −5.76448951883661492365788743004, −4.83573078868540720277467655847, −3.21360583178032056055104710293, −1.86210278350056807746831520728, −0.61769729530917183971331212638, 1.97693843730179872237275026970, 3.67672625531474787151937675337, 4.54947189772269306359297500816, 5.72024866552129339781275708795, 6.78731989516611626915309197395, 7.71944923350351475040003537215, 8.760526811438914817422823191928, 9.221045022390559970484117932414, 10.59219783107318702153330863387, 10.95398022586650207702629642657

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