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Properties

Label 2-546-273.146-c1-0-11
Degree $2$
Conductor $546$
Sign $-0.176 - 0.984i$
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 + (0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + 3.46·5-s + (−1.5 − 0.866i)6-s + (0.5 + 2.59i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−2.99 + 1.73i)10-s + 1.73·12-s + (−3.5 + 0.866i)13-s + (−1.73 − 2i)14-s + (2.99 + 5.19i)15-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s − 3i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + 1.54·5-s + (−0.612 − 0.353i)6-s + (0.188 + 0.981i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.948 + 0.547i)10-s + 0.499·12-s + (−0.970 + 0.240i)13-s + (−0.462 − 0.534i)14-s + (0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + (0.210 − 0.363i)17-s − 0.707i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.176 - 0.984i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01963 + 1.21857i\)
\(L(\frac12)\) \(\approx\) \(1.01963 + 1.21857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
13 \( 1 + (3.5 - 0.866i)T \)
good5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.866 + 1.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.19 + 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 + 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.9 - 7.5i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (7.79 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15 - 8.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.54507736664622873519811484186, −10.00157847266662845951578550339, −9.166137375182227261810257608064, −8.857629340978120614309195844748, −7.63965228160706084731862686178, −6.38556416898730539464868350404, −5.46333828162002753260487604304, −4.80520563034396675712180874344, −2.81006683163026639849991467091, −2.02662889901884554633845681718, 1.14280338685557149337617833506, 2.16583274808885723291888675758, 3.26784755989072678240482359509, 4.99813462552597712918966364300, 6.29280873905890804867195163966, 7.03171658278538309707453315981, 7.913404351317510659836037220826, 8.858820249398102415042936491161, 9.800556338728787068138559852898, 10.23334184676300509473401872580

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