Properties

Label 2-546-1.1-c7-0-67
Degree $2$
Conductor $546$
Sign $-1$
Analytic cond. $170.562$
Root an. cond. $13.0599$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 27·3-s + 64·4-s − 135·5-s − 216·6-s + 343·7-s + 512·8-s + 729·9-s − 1.08e3·10-s + 4.19e3·11-s − 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 3.64e3·15-s + 4.09e3·16-s − 1.27e4·17-s + 5.83e3·18-s − 2.82e4·19-s − 8.64e3·20-s − 9.26e3·21-s + 3.35e4·22-s − 5.70e4·23-s − 1.38e4·24-s − 5.99e4·25-s + 1.75e4·26-s − 1.96e4·27-s + 2.19e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.482·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.341·10-s + 0.950·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.278·15-s + 1/4·16-s − 0.628·17-s + 0.235·18-s − 0.943·19-s − 0.241·20-s − 0.218·21-s + 0.672·22-s − 0.977·23-s − 0.204·24-s − 0.766·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(170.562\)
Root analytic conductor: \(13.0599\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{3} T \)
3 \( 1 + p^{3} T \)
7 \( 1 - p^{3} T \)
13 \( 1 - p^{3} T \)
good5 \( 1 + 27 p T + p^{7} T^{2} \)
11 \( 1 - 4197 T + p^{7} T^{2} \)
17 \( 1 + 12735 T + p^{7} T^{2} \)
19 \( 1 + 28213 T + p^{7} T^{2} \)
23 \( 1 + 57039 T + p^{7} T^{2} \)
29 \( 1 + 10269 T + p^{7} T^{2} \)
31 \( 1 - 98276 T + p^{7} T^{2} \)
37 \( 1 + 352033 T + p^{7} T^{2} \)
41 \( 1 - 473172 T + p^{7} T^{2} \)
43 \( 1 - 891395 T + p^{7} T^{2} \)
47 \( 1 - 684984 T + p^{7} T^{2} \)
53 \( 1 - 271002 T + p^{7} T^{2} \)
59 \( 1 + 954024 T + p^{7} T^{2} \)
61 \( 1 - 3197159 T + p^{7} T^{2} \)
67 \( 1 + 2902018 T + p^{7} T^{2} \)
71 \( 1 + 4599768 T + p^{7} T^{2} \)
73 \( 1 - 1277111 T + p^{7} T^{2} \)
79 \( 1 + 1928494 T + p^{7} T^{2} \)
83 \( 1 + 355674 T + p^{7} T^{2} \)
89 \( 1 + 217854 T + p^{7} T^{2} \)
97 \( 1 + 4400758 T + p^{7} T^{2} \)
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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

−9.214274797297948456064618775694, −8.213663247073090905588359620909, −7.21830599901927250170162794147, −6.33534133821146902410671781088, −5.59675108818321936155218270203, −4.24319206421528847620621857468, −4.02066188337132426350134019025, −2.39825111046784900767692870102, −1.29806096478780927900362405597, 0, 1.29806096478780927900362405597, 2.39825111046784900767692870102, 4.02066188337132426350134019025, 4.24319206421528847620621857468, 5.59675108818321936155218270203, 6.33534133821146902410671781088, 7.21830599901927250170162794147, 8.213663247073090905588359620909, 9.214274797297948456064618775694

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