↑

Properties

Label 2-462-11.9-c1-0-8
Degree $2$
Conductor $462$
Sign $-0.190 + 0.981i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.0895 + 0.275i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 0.289·10-s + (3.06 − 1.26i)11-s + 0.999·12-s + (1.16 − 3.58i)13-s + (−0.809 + 0.587i)14-s + (−0.234 − 0.170i)15-s + (0.309 + 0.951i)16-s + (−2.03 − 6.25i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.0400 + 0.123i)5-s + (0.126 + 0.388i)6-s + (−0.305 − 0.222i)7-s + (−0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + 0.0916·10-s + (0.924 − 0.380i)11-s + 0.288·12-s + (0.322 − 0.993i)13-s + (−0.216 + 0.157i)14-s + (−0.0605 − 0.0439i)15-s + (0.0772 + 0.237i)16-s + (−0.492 − 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746254 - 0.905150i\)
\(L(\frac12)\) \(\approx\) \(0.746254 - 0.905150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-3.06 + 1.26i)T \)
good5 \( 1 + (-0.0895 - 0.275i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.16 + 3.58i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.03 + 6.25i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.879 - 0.638i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.94T + 23T^{2} \)
29 \( 1 + (5.54 + 4.03i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.41 + 4.34i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.00 - 2.90i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.36 - 6.80i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.40T + 43T^{2} \)
47 \( 1 + (-0.774 + 0.562i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.608 + 1.87i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.17 + 3.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.537 + 1.65i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.92T + 67T^{2} \)
71 \( 1 + (-1.72 - 5.32i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.02 - 2.19i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.26 - 10.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.87 - 14.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (1.10 - 3.38i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.06102179986870059076262965309, −9.936288126360427167825092548660, −9.372545232387708941336296357951, −8.253462656283318076147506448866, −6.88801341239455037565441222457, −5.97802644577082442791424422760, −4.89722075626651762947835647334, −3.84599303749077107244115526535, −2.75445705703679469998216964324, −0.76812886309882771550959494721, 1.69708343738000190971245656694, 3.67400440816457129355981211306, 4.68229232468318254222525898269, 5.88492989770994286078687516137, 6.62547287552656716617238554929, 7.31891498517362838371641413639, 8.745157344525899135657407560791, 9.128434983951688807516122353525, 10.50473077950467631614530378959, 11.38378412204248356893637602788

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