Properties

Label 2-546-1.1-c3-0-35
Degree $2$
Conductor $546$
Sign $-1$
Analytic cond. $32.2150$
Root an. cond. $5.67582$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s − 12·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 24·10-s − 50·11-s + 12·12-s − 13·13-s + 14·14-s − 36·15-s + 16·16-s − 58·17-s + 18·18-s − 40·19-s − 48·20-s + 21·21-s − 100·22-s − 64·23-s + 24·24-s + 19·25-s − 26·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.07·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.758·10-s − 1.37·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.619·15-s + 1/4·16-s − 0.827·17-s + 0.235·18-s − 0.482·19-s − 0.536·20-s + 0.218·21-s − 0.969·22-s − 0.580·23-s + 0.204·24-s + 0.151·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 - p T \)
7 \( 1 - p T \)
13 \( 1 + p T \)
good5 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 + 50 T + p^{3} T^{2} \)
17 \( 1 + 58 T + p^{3} T^{2} \)
19 \( 1 + 40 T + p^{3} T^{2} \)
23 \( 1 + 64 T + p^{3} T^{2} \)
29 \( 1 + 110 T + p^{3} T^{2} \)
31 \( 1 - 4 p T + p^{3} T^{2} \)
37 \( 1 + 50 T + p^{3} T^{2} \)
41 \( 1 - 84 T + p^{3} T^{2} \)
43 \( 1 + 12 T + p^{3} T^{2} \)
47 \( 1 + 82 T + p^{3} T^{2} \)
53 \( 1 + 442 T + p^{3} T^{2} \)
59 \( 1 + 618 T + p^{3} T^{2} \)
61 \( 1 + 278 T + p^{3} T^{2} \)
67 \( 1 - 20 T + p^{3} T^{2} \)
71 \( 1 + 390 T + p^{3} T^{2} \)
73 \( 1 + 2 T + p^{3} T^{2} \)
79 \( 1 + 680 T + p^{3} T^{2} \)
83 \( 1 - 322 T + p^{3} T^{2} \)
89 \( 1 - 968 T + p^{3} T^{2} \)
97 \( 1 - 1022 T + p^{3} T^{2} \)
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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

−10.16522992023059115045692566164, −8.857169863778516554000620288379, −7.87800248769478964902604992883, −7.52894366816861688497085947805, −6.24288124345212172400088715348, −4.93817934626293742262588912438, −4.22765809228520415877980760274, −3.12625116406523443539200642280, −2.06257327171424805485191744490, 0, 2.06257327171424805485191744490, 3.12625116406523443539200642280, 4.22765809228520415877980760274, 4.93817934626293742262588912438, 6.24288124345212172400088715348, 7.52894366816861688497085947805, 7.87800248769478964902604992883, 8.857169863778516554000620288379, 10.16522992023059115045692566164

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