Properties

Label 2-546-1.1-c5-0-52
Degree $2$
Conductor $546$
Sign $-1$
Analytic cond. $87.5695$
Root an. cond. $9.35786$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s + 33·5-s − 36·6-s + 49·7-s + 64·8-s + 81·9-s + 132·10-s − 94·11-s − 144·12-s − 169·13-s + 196·14-s − 297·15-s + 256·16-s − 1.11e3·17-s + 324·18-s − 1.90e3·19-s + 528·20-s − 441·21-s − 376·22-s + 961·23-s − 576·24-s − 2.03e3·25-s − 676·26-s − 729·27-s + 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.590·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.417·10-s − 0.234·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.340·15-s + 1/4·16-s − 0.933·17-s + 0.235·18-s − 1.21·19-s + 0.295·20-s − 0.218·21-s − 0.165·22-s + 0.378·23-s − 0.204·24-s − 0.651·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(87.5695\)
Root analytic conductor: \(9.35786\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{546} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 546,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 + p^{2} T \)
7 \( 1 - p^{2} T \)
13 \( 1 + p^{2} T \)
good5 \( 1 - 33 T + p^{5} T^{2} \)
11 \( 1 + 94 T + p^{5} T^{2} \)
17 \( 1 + 1112 T + p^{5} T^{2} \)
19 \( 1 + 1909 T + p^{5} T^{2} \)
23 \( 1 - 961 T + p^{5} T^{2} \)
29 \( 1 + 6337 T + p^{5} T^{2} \)
31 \( 1 - 7015 T + p^{5} T^{2} \)
37 \( 1 - 928 T + p^{5} T^{2} \)
41 \( 1 - 11442 T + p^{5} T^{2} \)
43 \( 1 + 12711 T + p^{5} T^{2} \)
47 \( 1 + 15107 T + p^{5} T^{2} \)
53 \( 1 - 18691 T + p^{5} T^{2} \)
59 \( 1 + 12360 T + p^{5} T^{2} \)
61 \( 1 - 14110 T + p^{5} T^{2} \)
67 \( 1 + 53746 T + p^{5} T^{2} \)
71 \( 1 + 47748 T + p^{5} T^{2} \)
73 \( 1 + 25301 T + p^{5} T^{2} \)
79 \( 1 + 5447 T + p^{5} T^{2} \)
83 \( 1 - 29393 T + p^{5} T^{2} \)
89 \( 1 + 77621 T + p^{5} T^{2} \)
97 \( 1 - 73607 T + p^{5} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.791986166736818930708542501070, −8.680324806852511866283287919743, −7.55946809212350755846172983022, −6.55229469626350225188951769932, −5.84540840289370594528512432424, −4.88635843304958650860313594006, −4.09290257510662530825531539691, −2.56124328774597524177449317981, −1.61913246233736735192058645368, 0, 1.61913246233736735192058645368, 2.56124328774597524177449317981, 4.09290257510662530825531539691, 4.88635843304958650860313594006, 5.84540840289370594528512432424, 6.55229469626350225188951769932, 7.55946809212350755846172983022, 8.680324806852511866283287919743, 9.791986166736818930708542501070

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