Properties

Label 2-546-273.194-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.0412 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.34 − 1.09i)3-s + (−0.499 − 0.866i)4-s + (−2.44 − 1.41i)5-s + (1.61 − 0.613i)6-s + (−2.60 − 0.449i)7-s + 0.999·8-s + (0.599 + 2.93i)9-s + (2.44 − 1.41i)10-s + (−2.97 − 5.15i)11-s + (−0.278 + 1.70i)12-s + (3.07 + 1.87i)13-s + (1.69 − 2.03i)14-s + (1.73 + 4.58i)15-s + (−0.5 + 0.866i)16-s + (2.83 + 4.91i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.774 − 0.632i)3-s + (−0.249 − 0.433i)4-s + (−1.09 − 0.632i)5-s + (0.661 − 0.250i)6-s + (−0.985 − 0.169i)7-s + 0.353·8-s + (0.199 + 0.979i)9-s + (0.774 − 0.447i)10-s + (−0.897 − 1.55i)11-s + (−0.0802 + 0.493i)12-s + (0.854 + 0.520i)13-s + (0.452 − 0.543i)14-s + (0.448 + 1.18i)15-s + (−0.125 + 0.216i)16-s + (0.688 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0412 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0412 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0412 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0412 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193106 + 0.201249i\)
\(L(\frac12)\) \(\approx\) \(0.193106 + 0.201249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (1.34 + 1.09i)T \)
7 \( 1 + (2.60 + 0.449i)T \)
13 \( 1 + (-3.07 - 1.87i)T \)
good5 \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.97 + 5.15i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.83 - 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.972 - 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 + 1.40i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.0658iT - 29T^{2} \)
31 \( 1 + (-4.39 - 7.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.160 - 0.0928i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.04iT - 41T^{2} \)
43 \( 1 - 3.48T + 43T^{2} \)
47 \( 1 + (-6.77 - 3.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.77 - 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.6 - 6.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.8 + 7.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.912 + 0.527i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.16T + 71T^{2} \)
73 \( 1 + (0.140 + 0.242i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.18 + 7.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.940iT - 83T^{2} \)
89 \( 1 + (-4.61 - 2.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.60T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.83869077341394067332529990036, −10.43503789767303161111820841450, −8.895990200898124620100036684980, −8.176080285270154100086209801227, −7.60788644035614138245027835066, −6.21090290794742534378900470722, −5.99358574861860811796174644590, −4.57675220586393563466422943902, −3.38560599476252005924413175756, −1.03617690013615144279484199972, 0.25318471166352226475804518413, 2.76319774902535778441375381943, 3.71206929574201103051688113205, 4.67177210605548732654824208594, 5.92408683802040251923016545469, 7.14250060767738884114483518365, 7.76033415774390678026033524004, 9.191295388354618032615682001153, 9.940608374800515556026267848111, 10.52790829772552007392649477519

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