Properties

Label 2-462-77.9-c1-0-9
Degree $2$
Conductor $462$
Sign $0.552 - 0.833i$
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  + (0.669 + 0.743i)2-s + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.282 + 0.0601i)5-s + (0.309 + 0.951i)6-s + (1.34 − 2.27i)7-s + (−0.809 + 0.587i)8-s + (0.669 + 0.743i)9-s + (0.144 + 0.250i)10-s + (3.27 + 0.532i)11-s + (−0.5 + 0.866i)12-s + (−0.973 + 2.99i)13-s + (2.59 − 0.524i)14-s + (0.233 + 0.170i)15-s + (−0.978 − 0.207i)16-s + (2.65 − 2.95i)17-s + ⋯
L(s)  = 1  + (0.473 + 0.525i)2-s + (0.527 + 0.234i)3-s + (−0.0522 + 0.497i)4-s + (0.126 + 0.0268i)5-s + (0.126 + 0.388i)6-s + (0.508 − 0.860i)7-s + (−0.286 + 0.207i)8-s + (0.223 + 0.247i)9-s + (0.0457 + 0.0792i)10-s + (0.987 + 0.160i)11-s + (−0.144 + 0.249i)12-s + (−0.269 + 0.830i)13-s + (0.693 − 0.140i)14-s + (0.0604 + 0.0438i)15-s + (−0.244 − 0.0519i)16-s + (0.644 − 0.715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01382 + 1.08072i\)
\(L(\frac12)\) \(\approx\) \(2.01382 + 1.08072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
7 \( 1 + (-1.34 + 2.27i)T \)
11 \( 1 + (-3.27 - 0.532i)T \)
good5 \( 1 + (-0.282 - 0.0601i)T + (4.56 + 2.03i)T^{2} \)
13 \( 1 + (0.973 - 2.99i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.65 + 2.95i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (-0.616 - 5.86i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-0.00853 + 0.0147i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.10 + 2.98i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.30 - 0.915i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-7.01 + 3.12i)T + (24.7 - 27.4i)T^{2} \)
41 \( 1 + (3.30 - 2.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.89T + 43T^{2} \)
47 \( 1 + (0.427 + 4.06i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-5.39 + 1.14i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-1.13 + 10.7i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-5.85 - 1.24i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-5.12 - 8.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.24 + 13.0i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.763 - 7.26i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (9.90 + 11.0i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-1.54 - 4.73i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-0.835 + 1.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.62 - 14.2i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.41622683063524937649762694337, −10.04695108551103565085681685756, −9.459357171042273448334118103998, −8.269837690984954968463373143609, −7.50417021072377538430058968826, −6.66607302876339282318535203641, −5.44318511211982160790494413606, −4.25320671593650508190611894707, −3.62636256332634953695860099690, −1.82569587512245668732920118024, 1.51446225368230892337279061740, 2.74818195005437080838103367060, 3.84249114107417210189692162874, 5.16241441796182907240293335382, 6.00832757720701685025491369147, 7.25022041776671003576387899776, 8.359621527811637891348339149202, 9.160410204542280855335343008342, 9.960395880059537049811292203883, 11.17831969155745252058211738317

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