Properties

Label 2-462-21.5-c1-0-21
Degree $2$
Conductor $462$
Sign $0.667 + 0.744i$
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  + (0.866 + 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s + (1.73 − 3i)5-s − 1.73i·6-s + (2 + 1.73i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (3 − 1.73i)10-s + (0.866 − 0.5i)11-s + (0.866 − 1.49i)12-s − 3.46i·13-s + (0.866 + 2.5i)14-s − 6·15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s + (0.774 − 1.34i)5-s − 0.707i·6-s + (0.755 + 0.654i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.948 − 0.547i)10-s + (0.261 − 0.150i)11-s + (0.250 − 0.433i)12-s − 0.960i·13-s + (0.231 + 0.668i)14-s − 1.54·15-s + (−0.125 + 0.216i)16-s + (−0.210 − 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82187 - 0.814120i\)
\(L(\frac12)\) \(\approx\) \(1.82187 - 0.814120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
7 \( 1 + (-2 - 1.73i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (0.866 + 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.06 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9 - 5.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.5iT - 97T^{2} \)
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

−11.39135174453451074882207438870, −10.05950086594889462282749100532, −8.682612786787120031946021037168, −8.316742650068183773271706853247, −7.11023929521331669129939663700, −5.91390324463607909486627665871, −5.38032588545960324396405292415, −4.62258253235883166671872217703, −2.54817334375893223770278762989, −1.25485179781589831187800653972, 1.93242226107785101022012739884, 3.37302914381018434139894033212, 4.33412679702544244760459956502, 5.33802346703479719624049578417, 6.48539917634026910444066643953, 6.98815092020727939280786477730, 8.710549140403202264873176760769, 9.869621554511272484706868965951, 10.43888782961734727242794869755, 11.10773251716136037417059808331

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