Properties

Label 2-462-11.5-c1-0-1
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

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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: $\chi_{462} (379, \cdot )$
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 \( 1 + (1.10 + 3.38i)T + (-78.4 + 57.0i)T^{2} \)
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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

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

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