Properties

Label 2-546-273.173-c1-0-27
Degree $2$
Conductor $546$
Sign $-0.678 + 0.734i$
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 + (−1.17 − 1.27i)3-s + (−0.499 + 0.866i)4-s + (0.870 + 0.502i)5-s + (−0.517 + 1.65i)6-s + (1.33 − 2.28i)7-s + 0.999·8-s + (−0.249 + 2.98i)9-s − 1.00i·10-s + 0.620·11-s + (1.69 − 0.378i)12-s + (1.14 − 3.41i)13-s + (−2.64 − 0.0151i)14-s + (−0.380 − 1.69i)15-s + (−0.5 − 0.866i)16-s + (0.171 − 0.296i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.677 − 0.735i)3-s + (−0.249 + 0.433i)4-s + (0.389 + 0.224i)5-s + (−0.211 + 0.674i)6-s + (0.504 − 0.863i)7-s + 0.353·8-s + (−0.0831 + 0.996i)9-s − 0.317i·10-s + 0.187·11-s + (0.487 − 0.109i)12-s + (0.316 − 0.948i)13-s + (−0.707 − 0.00404i)14-s + (−0.0981 − 0.438i)15-s + (−0.125 − 0.216i)16-s + (0.0415 − 0.0720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394460 - 0.900886i\)
\(L(\frac12)\) \(\approx\) \(0.394460 - 0.900886i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 1.27i)T \)
7 \( 1 + (-1.33 + 2.28i)T \)
13 \( 1 + (-1.14 + 3.41i)T \)
good5 \( 1 + (-0.870 - 0.502i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.620T + 11T^{2} \)
17 \( 1 + (-0.171 + 0.296i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 4.33T + 19T^{2} \)
23 \( 1 + (-2.44 + 1.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.23 + 4.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.76 - 2.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.47 + 4.31i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.602 + 1.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0442 - 0.0255i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.15 + 2.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.67 - 1.54i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.68iT - 61T^{2} \)
67 \( 1 + 5.48iT - 67T^{2} \)
71 \( 1 + (0.621 + 1.07i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.46 - 7.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.458 + 0.793i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + (-3.11 + 1.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.10 - 8.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.48431175754542947111698788072, −10.00933851735795309049220527935, −8.628371098508247969719524322891, −7.71419581688254179277770962336, −7.04383598255988945035376804846, −5.85713562497641217780839014104, −4.88023457152607828214620071662, −3.47770790341713073370323522005, −1.99909602450886702715694302829, −0.75446882387470049111612512599, 1.59988617314825328028562975807, 3.61440100076441367259617170192, 4.97048698749565753808621089785, 5.50121040876197760466944573354, 6.42045756202627678933379274896, 7.47760434617031168666265714886, 8.825933801007747537437034209410, 9.229410360778372635217889846036, 10.01841543730764584772906320568, 11.25322782048277876387430267520

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