Properties

Label 2-462-1.1-c7-0-30
Degree $2$
Conductor $462$
Sign $1$
Analytic cond. $144.321$
Root an. cond. $12.0134$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 27·3-s + 64·4-s + 182·5-s + 216·6-s + 343·7-s − 512·8-s + 729·9-s − 1.45e3·10-s + 1.33e3·11-s − 1.72e3·12-s + 1.37e4·13-s − 2.74e3·14-s − 4.91e3·15-s + 4.09e3·16-s + 1.47e4·17-s − 5.83e3·18-s + 3.45e4·19-s + 1.16e4·20-s − 9.26e3·21-s − 1.06e4·22-s + 4.98e4·23-s + 1.38e4·24-s − 4.50e4·25-s − 1.10e5·26-s − 1.96e4·27-s + 2.19e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.651·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.460·10-s + 0.301·11-s − 0.288·12-s + 1.73·13-s − 0.267·14-s − 0.375·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.15·19-s + 0.325·20-s − 0.218·21-s − 0.213·22-s + 0.853·23-s + 0.204·24-s − 0.576·25-s − 1.22·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(144.321\)
Root analytic conductor: \(12.0134\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: $\chi_{462} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.255891734\)
\(L(\frac12)\) \(\approx\) \(2.255891734\)
\(L(\frac{9}{2})\) 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 \( 1 - p^{3} T \)
11 \( 1 - p^{3} T \)
good5 \( 1 - 182 T + p^{7} T^{2} \)
13 \( 1 - 13758 T + p^{7} T^{2} \)
17 \( 1 - 14738 T + p^{7} T^{2} \)
19 \( 1 - 34516 T + p^{7} T^{2} \)
23 \( 1 - 49816 T + p^{7} T^{2} \)
29 \( 1 - 141174 T + p^{7} T^{2} \)
31 \( 1 + 86112 T + p^{7} T^{2} \)
37 \( 1 - 535038 T + p^{7} T^{2} \)
41 \( 1 - 708282 T + p^{7} T^{2} \)
43 \( 1 - 162292 T + p^{7} T^{2} \)
47 \( 1 - 397296 T + p^{7} T^{2} \)
53 \( 1 - 224798 T + p^{7} T^{2} \)
59 \( 1 - 694668 T + p^{7} T^{2} \)
61 \( 1 - 50110 T + p^{7} T^{2} \)
67 \( 1 + 3228404 T + p^{7} T^{2} \)
71 \( 1 + 1871496 T + p^{7} T^{2} \)
73 \( 1 + 937542 T + p^{7} T^{2} \)
79 \( 1 - 1266032 T + p^{7} T^{2} \)
83 \( 1 + 2545004 T + p^{7} T^{2} \)
89 \( 1 + 9909830 T + p^{7} T^{2} \)
97 \( 1 + 14010830 T + p^{7} T^{2} \)
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

−9.829405920894000291691912827216, −9.133856144548257560843608906051, −8.135011519794218559051469688727, −7.19682608063025506691719511870, −6.07322749124437834456808094019, −5.59004586298853140879180687172, −4.14534935229603494998595425631, −2.84316919769323409149101251497, −1.36196405662108944880561820494, −0.922494077665475583469570564492, 0.922494077665475583469570564492, 1.36196405662108944880561820494, 2.84316919769323409149101251497, 4.14534935229603494998595425631, 5.59004586298853140879180687172, 6.07322749124437834456808094019, 7.19682608063025506691719511870, 8.135011519794218559051469688727, 9.133856144548257560843608906051, 9.829405920894000291691912827216

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