Properties

Label 2-546-1.1-c1-0-4
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 + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 3·11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 5·17-s − 18-s + 19-s + 20-s − 21-s − 3·22-s + 3·23-s − 24-s − 4·25-s + 26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s − 0.639·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-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: 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

\(L(1)\) \(\approx\) \(1.427084885\)
\(L(\frac12)\) \(\approx\) \(1.427084885\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−10.37222580911320193962631624501, −9.879081886329772384809709299329, −9.088442719362802588356057027476, −8.303660229476383551053190981619, −7.28059517613580383361960117880, −6.48543346802406488839429568576, −5.35771852379483352074220354059, −3.81468304746563960569702394948, −2.71288319524339209617944628177, −1.30613462142742324235177250713, 1.30613462142742324235177250713, 2.71288319524339209617944628177, 3.81468304746563960569702394948, 5.35771852379483352074220354059, 6.48543346802406488839429568576, 7.28059517613580383361960117880, 8.303660229476383551053190981619, 9.088442719362802588356057027476, 9.879081886329772384809709299329, 10.37222580911320193962631624501

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