↑

Properties

Label 1-497-497.433-r0-0-0
Degree $1$
Conductor $497$
Sign $0.995 + 0.0914i$
Analytic cond. $2.30805$
Root an. cond. $2.30805$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.858 + 0.512i)2-s + (0.691 − 0.722i)3-s + (0.473 + 0.880i)4-s + (0.809 − 0.587i)5-s + (0.963 − 0.266i)6-s + (−0.0448 + 0.998i)8-s + (−0.0448 − 0.998i)9-s + (0.995 − 0.0896i)10-s + (−0.936 + 0.351i)11-s + (0.963 + 0.266i)12-s + (0.936 + 0.351i)13-s + (0.134 − 0.990i)15-s + (−0.550 + 0.834i)16-s + (0.309 + 0.951i)17-s + (0.473 − 0.880i)18-s + (−0.134 − 0.990i)19-s + ⋯
L(s)  = 1  + (0.858 + 0.512i)2-s + (0.691 − 0.722i)3-s + (0.473 + 0.880i)4-s + (0.809 − 0.587i)5-s + (0.963 − 0.266i)6-s + (−0.0448 + 0.998i)8-s + (−0.0448 − 0.998i)9-s + (0.995 − 0.0896i)10-s + (−0.936 + 0.351i)11-s + (0.963 + 0.266i)12-s + (0.936 + 0.351i)13-s + (0.134 − 0.990i)15-s + (−0.550 + 0.834i)16-s + (0.309 + 0.951i)17-s + (0.473 − 0.880i)18-s + (−0.134 − 0.990i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(497\)    =    \(7 \cdot 71\)
Sign: $0.995 + 0.0914i$
Analytic conductor: \(2.30805\)
Root analytic conductor: \(2.30805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{497} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 497,\ (0:\ ),\ 0.995 + 0.0914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.159706610 + 0.1447102758i\)
\(L(\frac12)\) \(\approx\) \(3.159706610 + 0.1447102758i\)
\(L(1)\) \(\approx\) \(2.269609937 + 0.1289489858i\)
\(L(1)\) \(\approx\) \(2.269609937 + 0.1289489858i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 \)
good2 \( 1 + (0.858 + 0.512i)T \)
3 \( 1 + (0.691 - 0.722i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.936 + 0.351i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.134 - 0.990i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (-0.550 - 0.834i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (-0.691 - 0.722i)T \)
53 \( 1 + (-0.473 + 0.880i)T \)
59 \( 1 + (-0.963 - 0.266i)T \)
61 \( 1 + (-0.995 + 0.0896i)T \)
67 \( 1 + (-0.473 - 0.880i)T \)
73 \( 1 + (-0.858 - 0.512i)T \)
79 \( 1 + (-0.0448 + 0.998i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (-0.473 + 0.880i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.235273174045731518025004616947, −22.77588076294922208185757598829, −21.77362324930988139291322041093, −21.00601654463049542847408160891, −20.79242416922782850591070695611, −19.67026478522692286735642932088, −18.66138961754144213990457560732, −18.08815550928641037180036375490, −16.30737135154201355809682712226, −15.91304384430342534706223210319, −14.6822110062124784035159178858, −14.25020218807537635840416636889, −13.43442795039000689098228659439, −12.685732601016393748050346442805, −11.1922016749825591644271939041, −10.53557718291149349773543035114, −9.95920814673962163398564756879, −8.880820563528993743273940915956, −7.66046274915803247509954980758, −6.322391635415695315034863348134, −5.46565019006399568043986936682, −4.561183848714685642286091001067, −3.200709688976054502622307155054, −2.84965976175887859635342339392, −1.59576866342753583655792583545, 1.53801978738223464740348276820, 2.465014091601139968968513029804, 3.561815764509578940500022033342, 4.76030759499724476513625863580, 5.816303630303895174047692830583, 6.5603965572428197195701374483, 7.643837322711265828917925091279, 8.45461144710656129978128761488, 9.28132492009397994903431864396, 10.66639847946912062935614295549, 11.957535133770124213763037354282, 12.80559279849589256167624387411, 13.45914521669063582647305008007, 13.86143017746878113890464161503, 15.093746624019846278803203159323, 15.678127690120758873452942992581, 16.90769779311870382591645485682, 17.62597817032441308113057868331, 18.44982701239533955573913249953, 19.67352558204180833282361951491, 20.573908350901209418994237338637, 21.15754705703011424965076437540, 21.837238891162443681704714746783, 23.26159389780559952708769915616, 23.76442455399032164197357284777

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