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Properties

Label 2-546-273.173-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.840 + 0.542i$
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 + (−0.428 + 1.67i)3-s + (−0.499 + 0.866i)4-s + (−1.58 − 0.917i)5-s + (−1.66 + 0.468i)6-s + (2.08 + 1.62i)7-s − 0.999·8-s + (−2.63 − 1.43i)9-s − 1.83i·10-s − 3.39·11-s + (−1.23 − 1.20i)12-s + (−3.07 + 1.88i)13-s + (−0.364 + 2.62i)14-s + (2.22 − 2.27i)15-s + (−0.5 − 0.866i)16-s + (−1.59 + 2.75i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.247 + 0.968i)3-s + (−0.249 + 0.433i)4-s + (−0.711 − 0.410i)5-s + (−0.680 + 0.191i)6-s + (0.788 + 0.614i)7-s − 0.353·8-s + (−0.877 − 0.478i)9-s − 0.580i·10-s − 1.02·11-s + (−0.357 − 0.349i)12-s + (−0.851 + 0.523i)13-s + (−0.0973 + 0.700i)14-s + (0.573 − 0.587i)15-s + (−0.125 − 0.216i)16-s + (−0.386 + 0.668i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.840 + 0.542i$
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: \((2,\ 546,\ (\ :1/2),\ -0.840 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197685 - 0.670777i\)
\(L(\frac12)\) \(\approx\) \(0.197685 - 0.670777i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.428 - 1.67i)T \)
7 \( 1 + (-2.08 - 1.62i)T \)
13 \( 1 + (3.07 - 1.88i)T \)
good5 \( 1 + (1.58 + 0.917i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.39T + 11T^{2} \)
17 \( 1 + (1.59 - 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.03T + 19T^{2} \)
23 \( 1 + (-2.14 + 1.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.81 + 1.04i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.14 - 5.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.436 - 0.251i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.35 - 2.51i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.528 - 0.916i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.0 + 5.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.81 - 3.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.37 - 3.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + 9.92iT - 67T^{2} \)
71 \( 1 + (-7.77 - 13.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.47 - 7.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.92 + 6.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + (4.61 - 2.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.79 - 8.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.36365753790200356706172375704, −10.51755603695272044041917031424, −9.432543684863755492528894958513, −8.442068101784676211167694387797, −8.031873772975897000400848725938, −6.68408853675020624201055001268, −5.47872495889953097107061981388, −4.79967975562907095502476347307, −4.10730180557185075949964373112, −2.61534015304727277928381029343, 0.35253278842585930059529036106, 2.08425947869254730663968371038, 3.16926420888325255371411163088, 4.63081615437599421746691513130, 5.41698166933260659041175002986, 6.77816661569973644838986492715, 7.67189535803646267253393564954, 8.103085246881181077481844796460, 9.590542052064870885321044261189, 10.78722675053181520058295194448

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