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Properties

Label 2-546-273.152-c1-0-15
Degree $2$
Conductor $546$
Sign $0.904 + 0.425i$
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 + (−1.67 − 0.437i)3-s + (0.499 − 0.866i)4-s + (1.40 − 2.42i)5-s + (1.67 − 0.459i)6-s + (1.42 + 2.22i)7-s + 0.999i·8-s + (2.61 + 1.46i)9-s + 2.80i·10-s − 2.61i·11-s + (−1.21 + 1.23i)12-s + (0.591 + 3.55i)13-s + (−2.34 − 1.21i)14-s + (−3.41 + 3.45i)15-s + (−0.5 − 0.866i)16-s + (−2.31 + 4.00i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.967 − 0.252i)3-s + (0.249 − 0.433i)4-s + (0.627 − 1.08i)5-s + (0.681 − 0.187i)6-s + (0.538 + 0.842i)7-s + 0.353i·8-s + (0.872 + 0.488i)9-s + 0.887i·10-s − 0.788i·11-s + (−0.351 + 0.355i)12-s + (0.164 + 0.986i)13-s + (−0.627 − 0.325i)14-s + (−0.881 + 0.893i)15-s + (−0.125 − 0.216i)16-s + (−0.561 + 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.904 + 0.425i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923700 - 0.206509i\)
\(L(\frac12)\) \(\approx\) \(0.923700 - 0.206509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.437i)T \)
7 \( 1 + (-1.42 - 2.22i)T \)
13 \( 1 + (-0.591 - 3.55i)T \)
good5 \( 1 + (-1.40 + 2.42i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.61iT - 11T^{2} \)
17 \( 1 + (2.31 - 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 6.81iT - 19T^{2} \)
23 \( 1 + (-4.59 + 2.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.91 - 3.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.96 + 3.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.497 - 0.860i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.10 - 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.67 + 6.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.77 + 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.16 + 5.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.53 - 2.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 - 9.93T + 67T^{2} \)
71 \( 1 + (-4.88 + 2.81i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.76 - 1.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.142 - 0.247i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.889T + 83T^{2} \)
89 \( 1 + (7.24 + 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.06 - 4.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.87799927484244633912573802287, −9.723086348228748564568138828161, −8.691302863691480253477267022382, −8.514870215705643797066214903067, −6.87797978014105141900047421109, −6.22717832923138185105772582549, −5.22677278982602334020764479419, −4.64064174975348053657102425192, −2.16851992395523006126471414853, −0.952658839162179719595821170062, 1.20766908611634085977598605394, 2.79727415993829522483553939472, 4.16741959896117817282279685821, 5.32527640981233277130514190187, 6.51598607108605030127740874750, 7.15026717879261280983660558559, 8.075654681451485639799203755894, 9.590651557919536544065633958917, 10.27073474408450245490810040817, 10.58044071863195767286349123281

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