Properties

Label 2-546-1.1-c1-0-6
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 + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 3·11-s + 12-s + 13-s − 14-s + 3·15-s + 16-s − 3·17-s − 18-s − 7·19-s + 3·20-s + 21-s − 3·22-s + 9·23-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 + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.60·19-s + 0.670·20-s + 0.218·21-s − 0.639·22-s + 1.87·23-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

\(L(1)\) \(\approx\) \(1.690956649\)
\(L(\frac12)\) \(\approx\) \(1.690956649\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−10.78600737924430448083487191522, −9.626606150401529001958201314809, −9.079799022404556504658251347602, −8.505529332786924267956180509594, −7.16011708962485875279239287487, −6.45501714883769400197534752234, −5.38516927860402892091834425845, −3.95823828965663710949182273306, −2.41262911451317421726904251873, −1.54544973372056719849266422970, 1.54544973372056719849266422970, 2.41262911451317421726904251873, 3.95823828965663710949182273306, 5.38516927860402892091834425845, 6.45501714883769400197534752234, 7.16011708962485875279239287487, 8.505529332786924267956180509594, 9.079799022404556504658251347602, 9.626606150401529001958201314809, 10.78600737924430448083487191522

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