Properties

Label 2-546-1.1-c1-0-8
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 − 0.561·5-s + 6-s − 7-s + 8-s + 9-s − 0.561·10-s + 2.56·11-s + 12-s + 13-s − 14-s − 0.561·15-s + 16-s + 5.68·17-s + 18-s + 7.68·19-s − 0.561·20-s − 21-s + 2.56·22-s − 1.43·23-s + 24-s − 4.68·25-s + 26-s + 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.251·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.177·10-s + 0.772·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.144·15-s + 0.250·16-s + 1.37·17-s + 0.235·18-s + 1.76·19-s − 0.125·20-s − 0.218·21-s + 0.546·22-s − 0.299·23-s + 0.204·24-s − 0.936·25-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

\(L(1)\) \(\approx\) \(2.630653470\)
\(L(\frac12)\) \(\approx\) \(2.630653470\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.561T + 5T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
17 \( 1 - 5.68T + 17T^{2} \)
19 \( 1 - 7.68T + 19T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 5.68T + 61T^{2} \)
67 \( 1 - 1.12T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 0.561T + 73T^{2} \)
79 \( 1 + 2.87T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−10.96102185939495039898716611437, −9.756888510714417580519514037867, −9.233370560023010947145845166715, −7.83825761795388458926986676872, −7.32618673380141590440426968804, −6.09386238561401144317148233861, −5.20605424187117081970844569536, −3.73522433295182470573736754474, −3.32218683883157689035440528794, −1.61386041996840783631232130153, 1.61386041996840783631232130153, 3.32218683883157689035440528794, 3.73522433295182470573736754474, 5.20605424187117081970844569536, 6.09386238561401144317148233861, 7.32618673380141590440426968804, 7.83825761795388458926986676872, 9.233370560023010947145845166715, 9.756888510714417580519514037867, 10.96102185939495039898716611437

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