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Properties

Label 2-546-273.17-c1-0-18
Degree $2$
Conductor $546$
Sign $0.687 - 0.725i$
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  + 2-s + (0.166 + 1.72i)3-s + 4-s + (1.41 − 0.815i)5-s + (0.166 + 1.72i)6-s + (2.62 + 0.292i)7-s + 8-s + (−2.94 + 0.575i)9-s + (1.41 − 0.815i)10-s + (1.03 + 1.79i)11-s + (0.166 + 1.72i)12-s + (−3.37 − 1.27i)13-s + (2.62 + 0.292i)14-s + (1.64 + 2.29i)15-s + 16-s + 3.05·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.0963 + 0.995i)3-s + 0.5·4-s + (0.631 − 0.364i)5-s + (0.0680 + 0.703i)6-s + (0.993 + 0.110i)7-s + 0.353·8-s + (−0.981 + 0.191i)9-s + (0.446 − 0.257i)10-s + (0.312 + 0.540i)11-s + (0.0481 + 0.497i)12-s + (−0.935 − 0.352i)13-s + (0.702 + 0.0782i)14-s + (0.424 + 0.593i)15-s + 0.250·16-s + 0.740·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.687 - 0.725i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.687 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41226 + 1.03753i\)
\(L(\frac12)\) \(\approx\) \(2.41226 + 1.03753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.166 - 1.72i)T \)
7 \( 1 + (-2.62 - 0.292i)T \)
13 \( 1 + (3.37 + 1.27i)T \)
good5 \( 1 + (-1.41 + 0.815i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.03 - 1.79i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + (-0.662 + 1.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.10iT - 23T^{2} \)
29 \( 1 + (2.79 + 1.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.28 - 5.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.93iT - 37T^{2} \)
41 \( 1 + (7.41 + 4.28i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.69 - 4.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.714 + 0.412i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.25 - 5.34i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.5iT - 59T^{2} \)
61 \( 1 + (13.4 + 7.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.83 - 1.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.59 + 7.95i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.43 + 2.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.24 + 5.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.85iT - 83T^{2} \)
89 \( 1 + 1.07iT - 89T^{2} \)
97 \( 1 + (4.58 + 7.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84817139829961481240168211595, −10.12068089546089996071545000459, −9.304105470554049790763165478865, −8.332398710813079146899190809488, −7.30329695548313035871852182173, −5.92733218496335757981718588967, −5.02368576418851024131081417732, −4.59979880348064902911983717927, −3.20280295427429580844575454052, −1.91455302357725584357571957564, 1.53802061969858942217742896979, 2.52825464039993991677210236430, 3.88099556385018048923855781406, 5.41183193843021727823305381885, 5.88475469412446391579683660140, 7.18008851970441151772229005749, 7.61994191160382796103380299542, 8.783986644919356036878843428847, 9.895901161831650271647123376033, 11.00604329818360255046673142244

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