Properties

Label 2-546-1.1-c1-0-10
Degree $2$
Conductor $546$
Sign $1$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s − 4·11-s + 12-s − 13-s + 14-s + 2·15-s + 16-s + 6·17-s + 18-s − 4·19-s + 2·20-s + 21-s − 4·22-s + 24-s − 25-s − 26-s + 27-s + 28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.954731260\)
\(L(\frac12)\) \(\approx\) \(2.954731260\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−10.66462432682540621042685892615, −10.06588058308824367876151942543, −9.102685498382662983764605131023, −7.927583890916278548349975461285, −7.31420502617006761020558985636, −5.87416482309586917103143441505, −5.34066639773140823665957604454, −4.08389192736738734823347258017, −2.81336235055681255128786870399, −1.85528951457630629269007018576, 1.85528951457630629269007018576, 2.81336235055681255128786870399, 4.08389192736738734823347258017, 5.34066639773140823665957604454, 5.87416482309586917103143441505, 7.31420502617006761020558985636, 7.927583890916278548349975461285, 9.102685498382662983764605131023, 10.06588058308824367876151942543, 10.66462432682540621042685892615

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