Properties

Label 2-546-1.1-c1-0-2
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 − 3.70·5-s − 6-s + 7-s + 8-s + 9-s − 3.70·10-s + 5.70·11-s − 12-s + 13-s + 14-s + 3.70·15-s + 16-s + 3.70·17-s + 18-s + 5.70·19-s − 3.70·20-s − 21-s + 5.70·22-s − 1.70·23-s − 24-s + 8.70·25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.65·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.17·10-s + 1.71·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.955·15-s + 0.250·16-s + 0.897·17-s + 0.235·18-s + 1.30·19-s − 0.827·20-s − 0.218·21-s + 1.21·22-s − 0.354·23-s − 0.204·24-s + 1.74·25-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

\(L(1)\) \(\approx\) \(1.649556773\)
\(L(\frac12)\) \(\approx\) \(1.649556773\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.70T + 5T^{2} \)
11 \( 1 - 5.70T + 11T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 + 9.40T + 41T^{2} \)
43 \( 1 - 9.10T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 7.70T + 61T^{2} \)
67 \( 1 + 7.40T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 7.70T + 73T^{2} \)
79 \( 1 + 3.40T + 79T^{2} \)
83 \( 1 + 0.596T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−11.43483643089544292602743656076, −10.15243844260323863445322529913, −8.953394879882585085534439410086, −7.81391489647087793893531511872, −7.19826094541585290985540074696, −6.17195708811421168737571013421, −5.03677385107277837904297481321, −4.02564892686347259075982140467, −3.44521090361969176928239126777, −1.18483450496188729564153348359, 1.18483450496188729564153348359, 3.44521090361969176928239126777, 4.02564892686347259075982140467, 5.03677385107277837904297481321, 6.17195708811421168737571013421, 7.19826094541585290985540074696, 7.81391489647087793893531511872, 8.953394879882585085534439410086, 10.15243844260323863445322529913, 11.43483643089544292602743656076

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