Properties

Label 2-546-1.1-c1-0-5
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 + 11-s − 12-s − 13-s − 14-s − 3·15-s + 16-s + 7·17-s + 18-s + 19-s + 3·20-s + 21-s + 22-s − 7·23-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 + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.229·19-s + 0.670·20-s + 0.218·21-s + 0.213·22-s − 1.45·23-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

\(L(1)\) \(\approx\) \(2.278357824\)
\(L(\frac12)\) \(\approx\) \(2.278357824\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−10.72405745965352481313791757742, −9.988795367837157057695788102202, −9.435514649834190793278551180591, −7.925758442786961590691428846639, −6.84627164270088746798526670414, −5.86366672737057092676215077124, −5.55921197766306254502353102551, −4.23861623857583401252172064502, −2.89830493976033687883443576173, −1.52673365627799806411662380976, 1.52673365627799806411662380976, 2.89830493976033687883443576173, 4.23861623857583401252172064502, 5.55921197766306254502353102551, 5.86366672737057092676215077124, 6.84627164270088746798526670414, 7.925758442786961590691428846639, 9.435514649834190793278551180591, 9.988795367837157057695788102202, 10.72405745965352481313791757742

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