↑

Properties

Label 2-546-273.257-c1-0-1
Degree $2$
Conductor $546$
Sign $0.0555 - 0.998i$
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 + (−0.436 − 1.67i)3-s + 4-s + (−0.567 − 0.327i)5-s + (0.436 + 1.67i)6-s + (−2.37 − 1.17i)7-s − 8-s + (−2.61 + 1.46i)9-s + (0.567 + 0.327i)10-s + (−1.54 + 2.67i)11-s + (−0.436 − 1.67i)12-s + (3.50 + 0.844i)13-s + (2.37 + 1.17i)14-s + (−0.301 + 1.09i)15-s + 16-s − 7.02·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.252 − 0.967i)3-s + 0.5·4-s + (−0.253 − 0.146i)5-s + (0.178 + 0.684i)6-s + (−0.896 − 0.442i)7-s − 0.353·8-s + (−0.872 + 0.487i)9-s + (0.179 + 0.103i)10-s + (−0.466 + 0.807i)11-s + (−0.126 − 0.483i)12-s + (0.972 + 0.234i)13-s + (0.634 + 0.312i)14-s + (−0.0778 + 0.282i)15-s + 0.250·16-s − 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0555 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0555 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0555 - 0.998i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0555 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.207244 + 0.196026i\)
\(L(\frac12)\) \(\approx\) \(0.207244 + 0.196026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.436 + 1.67i)T \)
7 \( 1 + (2.37 + 1.17i)T \)
13 \( 1 + (-3.50 - 0.844i)T \)
good5 \( 1 + (0.567 + 0.327i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.54 - 2.67i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.02T + 17T^{2} \)
19 \( 1 + (-3.25 - 5.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.43iT - 23T^{2} \)
29 \( 1 + (-3.89 + 2.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.26 - 3.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.96iT - 37T^{2} \)
41 \( 1 + (7.52 - 4.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0380 - 0.0658i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.04 - 4.64i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.68 - 5.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.07iT - 59T^{2} \)
61 \( 1 + (13.2 - 7.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 1.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.469 - 0.813i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.44 + 9.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.40 - 2.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.24iT - 83T^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + (8.57 - 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.91043476600451017125458265631, −10.20680368149140731664792651761, −9.153428401415813254825205907486, −8.249044467384189824723745227588, −7.45296764755119567789480748002, −6.61200607030017512732459877328, −5.93593908122777629196639440821, −4.31944504079909517875701564252, −2.83533389763802231420388216586, −1.45707706366784642410043206249, 0.21453519105682776071236773058, 2.74065540629921638328582854452, 3.60041484305897093313585273786, 5.02338981148252281143041134406, 6.10774563012341480736926570531, 6.82584979562007959359738343306, 8.314786888037510357031071844586, 8.937596434343547847641317788721, 9.605666156196946678920532291753, 10.67582984761288101178524105685

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