↑

Properties

Label 2-546-273.17-c1-0-19
Degree $2$
Conductor $546$
Sign $0.970 - 0.242i$
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  + 2-s + (1.73 + 0.0694i)3-s + 4-s + (−2.64 + 1.52i)5-s + (1.73 + 0.0694i)6-s + (0.969 − 2.46i)7-s + 8-s + (2.99 + 0.240i)9-s + (−2.64 + 1.52i)10-s + (1.13 + 1.96i)11-s + (1.73 + 0.0694i)12-s + (3.11 + 1.81i)13-s + (0.969 − 2.46i)14-s + (−4.69 + 2.46i)15-s + 16-s + 3.59·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.999 + 0.0400i)3-s + 0.5·4-s + (−1.18 + 0.684i)5-s + (0.706 + 0.0283i)6-s + (0.366 − 0.930i)7-s + 0.353·8-s + (0.996 + 0.0801i)9-s + (−0.837 + 0.483i)10-s + (0.341 + 0.591i)11-s + (0.499 + 0.0200i)12-s + (0.864 + 0.502i)13-s + (0.259 − 0.657i)14-s + (−1.21 + 0.635i)15-s + 0.250·16-s + 0.872·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74275 + 0.337730i\)
\(L(\frac12)\) \(\approx\) \(2.74275 + 0.337730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.73 - 0.0694i)T \)
7 \( 1 + (-0.969 + 2.46i)T \)
13 \( 1 + (-3.11 - 1.81i)T \)
good5 \( 1 + (2.64 - 1.52i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.13 - 1.96i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
19 \( 1 + (0.413 - 0.716i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.62iT - 23T^{2} \)
29 \( 1 + (7.61 + 4.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.78 - 8.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.34iT - 37T^{2} \)
41 \( 1 + (7.03 + 4.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.70 + 8.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.73 - 1.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.44 + 0.834i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (9.14 + 5.28i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.51 + 2.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.81 + 6.60i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.74 + 9.94i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.91 - 5.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.79iT - 83T^{2} \)
89 \( 1 + 8.34iT - 89T^{2} \)
97 \( 1 + (-7.99 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.82746300476795542634443115991, −10.22152756554091966372881382067, −8.906563352384482825310965964816, −7.890735927987722418275913905581, −7.30706936194671278695449065820, −6.58230690760226174282068890859, −4.80481058278080075500579597144, −3.77728236487891434330175617398, −3.48253367473951695672178446935, −1.76144323168996725442987305472, 1.56374374974124948661499900062, 3.23814378554747189021935939595, 3.80130467393816869837874691391, 5.00674591607441024651549151266, 5.98910570115479077983056647572, 7.46052717650032502779079501529, 8.098720607354761586965187442175, 8.758055536365429342397049643362, 9.679127882026955886669202916048, 11.21990152976125651334273938512

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