Properties

Label 2-546-1.1-c1-0-0
Degree $2$
Conductor $546$
Sign $1$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 4.27·5-s + 6-s + 7-s − 8-s + 9-s + 4.27·10-s − 2.27·11-s − 12-s − 13-s − 14-s + 4.27·15-s + 16-s + 0.274·17-s − 18-s − 2.27·19-s − 4.27·20-s − 21-s + 2.27·22-s + 2.27·23-s + 24-s + 13.2·25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.91·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.35·10-s − 0.685·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1.10·15-s + 0.250·16-s + 0.0666·17-s − 0.235·18-s − 0.521·19-s − 0.955·20-s − 0.218·21-s + 0.485·22-s + 0.474·23-s + 0.204·24-s + 2.65·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-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: no
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5044367619\)
\(L(\frac12)\) \(\approx\) \(0.5044367619\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.27T + 5T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
17 \( 1 - 0.274T + 17T^{2} \)
19 \( 1 + 2.27T + 19T^{2} \)
23 \( 1 - 2.27T + 23T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 - 6.54T + 41T^{2} \)
43 \( 1 + 2.27T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + 4.54T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−10.78868508975187744268244702582, −10.21964007909320570154373489479, −8.742751029092682789085905086951, −8.118377903121081729655039409972, −7.39387180731048144549132281965, −6.56221015366234753717816573741, −5.05266373541693740873949136430, −4.19611378843197887720672721530, −2.83488970863995323177745930789, −0.71120220211023209380963073748, 0.71120220211023209380963073748, 2.83488970863995323177745930789, 4.19611378843197887720672721530, 5.05266373541693740873949136430, 6.56221015366234753717816573741, 7.39387180731048144549132281965, 8.118377903121081729655039409972, 8.742751029092682789085905086951, 10.21964007909320570154373489479, 10.78868508975187744268244702582

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