↑

Properties

Label 2-546-21.17-c1-0-7
Degree $2$
Conductor $546$
Sign $-0.268 - 0.963i$
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 + (1.11 + 1.32i)3-s + (0.499 − 0.866i)4-s + (0.737 + 1.27i)5-s + (−1.62 − 0.591i)6-s + (2.24 + 1.39i)7-s + 0.999i·8-s + (−0.517 + 2.95i)9-s + (−1.27 − 0.737i)10-s + (0.616 + 0.356i)11-s + (1.70 − 0.301i)12-s − i·13-s + (−2.64 − 0.0805i)14-s + (−0.872 + 2.40i)15-s + (−0.5 − 0.866i)16-s + (0.629 − 1.09i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.643 + 0.765i)3-s + (0.249 − 0.433i)4-s + (0.329 + 0.571i)5-s + (−0.664 − 0.241i)6-s + (0.850 + 0.526i)7-s + 0.353i·8-s + (−0.172 + 0.984i)9-s + (−0.404 − 0.233i)10-s + (0.186 + 0.107i)11-s + (0.492 − 0.0870i)12-s − 0.277i·13-s + (−0.706 − 0.0215i)14-s + (−0.225 + 0.620i)15-s + (−0.125 − 0.216i)16-s + (0.152 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.268 - 0.963i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913814 + 1.20274i\)
\(L(\frac12)\) \(\approx\) \(0.913814 + 1.20274i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (-2.24 - 1.39i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-0.737 - 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.616 - 0.356i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.629 + 1.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 - 1.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.87 + 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.97iT - 29T^{2} \)
31 \( 1 + (1.42 + 0.824i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.738 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.74T + 41T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 + (-1.02 - 1.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.4 + 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.88 + 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.61 - 2.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 - 6.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.81iT - 71T^{2} \)
73 \( 1 + (1.31 + 0.760i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (-1.87 - 3.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.98iT - 97T^{2} \)
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

−10.80810444124660427342800150950, −10.08067418497695662421035085563, −9.253047196614622150526751380790, −8.448651364623158189461841481363, −7.77431524502299630007010798641, −6.62525527434611896024949890007, −5.48520362609591937105130360259, −4.55827979122301465062890733408, −3.05316998860173171005174296700, −1.94477230526401543301509313693, 1.09702911951868143355183528174, 2.05180604836957559433878513556, 3.49783460766130639525657360598, 4.75656677885824109373225273134, 6.18599278115850000102655356422, 7.25760314641934997718906503539, 7.961032374037453247529693934480, 8.822541845768557667489171076008, 9.380990376637862595240627365491, 10.54554802326411455687479603370

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