Properties

Label 2-546-273.185-c1-0-21
Degree $2$
Conductor $546$
Sign $0.985 - 0.170i$
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  + (0.866 + 0.5i)2-s + (0.0864 − 1.72i)3-s + (0.499 + 0.866i)4-s + (0.886 + 1.53i)5-s + (0.939 − 1.45i)6-s + (2.61 + 0.424i)7-s + 0.999i·8-s + (−2.98 − 0.299i)9-s + 1.77i·10-s + 0.893i·11-s + (1.54 − 0.790i)12-s + (1.84 − 3.09i)13-s + (2.04 + 1.67i)14-s + (2.73 − 1.40i)15-s + (−0.5 + 0.866i)16-s + (3.74 + 6.49i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.0499 − 0.998i)3-s + (0.249 + 0.433i)4-s + (0.396 + 0.686i)5-s + (0.383 − 0.593i)6-s + (0.987 + 0.160i)7-s + 0.353i·8-s + (−0.995 − 0.0997i)9-s + 0.560i·10-s + 0.269i·11-s + (0.444 − 0.228i)12-s + (0.511 − 0.859i)13-s + (0.547 + 0.447i)14-s + (0.705 − 0.361i)15-s + (−0.125 + 0.216i)16-s + (0.909 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.985 - 0.170i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.985 - 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37504 + 0.203493i\)
\(L(\frac12)\) \(\approx\) \(2.37504 + 0.203493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0864 + 1.72i)T \)
7 \( 1 + (-2.61 - 0.424i)T \)
13 \( 1 + (-1.84 + 3.09i)T \)
good5 \( 1 + (-0.886 - 1.53i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.893iT - 11T^{2} \)
17 \( 1 + (-3.74 - 6.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.805iT - 19T^{2} \)
23 \( 1 + (4.72 + 2.72i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.62 + 4.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.20 + 4.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.84 - 3.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.93 + 3.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.19 - 7.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.72 + 1.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.67 - 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.21iT - 61T^{2} \)
67 \( 1 + 7.98T + 67T^{2} \)
71 \( 1 + (1.97 + 1.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.54 + 4.35i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.40 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (7.75 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.24 + 2.45i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.86540559090514951180974636980, −10.26811302562113804786291616034, −8.538205104543032767791700752551, −8.041608691800966214082323143707, −7.19471837767829794694797678430, −6.01779357173933287174877392488, −5.71844558524518018993688732635, −4.13574706998489531428173945913, −2.79834626958663943849847452524, −1.67578979184530063411246195419, 1.48251520922200326701049884780, 3.07759167176368008202083128735, 4.22515491853418900479551038151, 5.08541942098735636474451729523, 5.60245860740816690275914664653, 7.09155438472694152202259877975, 8.404936893578403634176458497460, 9.156644504419499429844897384088, 9.941843071830049950545844867238, 10.92257071442978125762588252612

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