Properties

Label 2-546-273.254-c1-0-31
Degree $2$
Conductor $546$
Sign $-0.314 + 0.949i$
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  + (−0.965 − 0.258i)2-s + (1.60 − 0.650i)3-s + (0.866 + 0.499i)4-s + (0.124 + 0.465i)5-s + (−1.71 + 0.213i)6-s + (−0.410 − 2.61i)7-s + (−0.707 − 0.707i)8-s + (2.15 − 2.08i)9-s − 0.481i·10-s + (−4.14 − 4.14i)11-s + (1.71 + 0.238i)12-s + (−3.55 − 0.592i)13-s + (−0.279 + 2.63i)14-s + (0.502 + 0.665i)15-s + (0.500 + 0.866i)16-s + (−0.233 + 0.404i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.926 − 0.375i)3-s + (0.433 + 0.249i)4-s + (0.0557 + 0.207i)5-s + (−0.701 + 0.0870i)6-s + (−0.155 − 0.987i)7-s + (−0.249 − 0.249i)8-s + (0.717 − 0.696i)9-s − 0.152i·10-s + (−1.25 − 1.25i)11-s + (0.495 + 0.0689i)12-s + (−0.986 − 0.164i)13-s + (−0.0746 + 0.703i)14-s + (0.129 + 0.171i)15-s + (0.125 + 0.216i)16-s + (−0.0565 + 0.0980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.314 + 0.949i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688849 - 0.954058i\)
\(L(\frac12)\) \(\approx\) \(0.688849 - 0.954058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.60 + 0.650i)T \)
7 \( 1 + (0.410 + 2.61i)T \)
13 \( 1 + (3.55 + 0.592i)T \)
good5 \( 1 + (-0.124 - 0.465i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.14 + 4.14i)T + 11iT^{2} \)
17 \( 1 + (0.233 - 0.404i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.05 + 1.05i)T + 19iT^{2} \)
23 \( 1 + (0.435 + 0.754i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.17 - 4.14i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.859 + 3.20i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.11 - 1.37i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.926 - 3.45i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.93 - 5.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.94 + 1.59i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.10 + 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.15 + 0.578i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 8.12T + 61T^{2} \)
67 \( 1 + (4.53 + 4.53i)T + 67iT^{2} \)
71 \( 1 + (-5.26 - 1.41i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-12.2 - 3.27i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.5 - 10.5i)T - 83iT^{2} \)
89 \( 1 + (-1.50 + 5.60i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.48 + 5.54i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.35304269465119820293762362066, −9.768444755072598892993391660262, −8.556712840939538787290776561744, −8.026844865856266295403747606796, −7.18674397383176875554828777687, −6.34153363066762076759685061276, −4.70055565104061768661500790439, −3.25648037376232752917919414739, −2.53460090300122080380194847007, −0.73755939104900453645192952614, 2.11265353577953866255930778849, 2.79772242616701550756424190993, 4.58745132729585574050284200928, 5.37248160472545124237757048740, 6.87888092616520407068207210367, 7.70753582190822594246362910603, 8.486354288101138297035639351554, 9.307835141425743793976660049955, 9.973997656133514036488620476331, 10.60448278834877347341785760016

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