↑

Properties

Label 1-473-473.269-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.956 - 0.292i$
Analytic cond. $2.19660$
Root an. cond. $2.19660$
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  + (−0.393 − 0.919i)2-s + (0.858 − 0.512i)3-s + (−0.691 + 0.722i)4-s + (−0.963 + 0.266i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (0.936 + 0.351i)8-s + (0.473 − 0.880i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (−0.0448 + 0.998i)13-s + (−0.995 + 0.0896i)14-s + (−0.691 + 0.722i)15-s + (−0.0448 − 0.998i)16-s + (−0.0448 − 0.998i)17-s + (−0.995 − 0.0896i)18-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (0.858 − 0.512i)3-s + (−0.691 + 0.722i)4-s + (−0.963 + 0.266i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (0.936 + 0.351i)8-s + (0.473 − 0.880i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (−0.0448 + 0.998i)13-s + (−0.995 + 0.0896i)14-s + (−0.691 + 0.722i)15-s + (−0.0448 − 0.998i)16-s + (−0.0448 − 0.998i)17-s + (−0.995 − 0.0896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.956 - 0.292i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1470263940 - 0.9847516700i\)
\(L(\frac12)\) \(\approx\) \(0.1470263940 - 0.9847516700i\)
\(L(1)\) \(\approx\) \(0.6648486815 - 0.6235393969i\)
\(L(1)\) \(\approx\) \(0.6648486815 - 0.6235393969i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.393 - 0.919i)T \)
3 \( 1 + (0.858 - 0.512i)T \)
5 \( 1 + (-0.963 + 0.266i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.0448 + 0.998i)T \)
17 \( 1 + (-0.0448 - 0.998i)T \)
19 \( 1 + (0.134 - 0.990i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (-0.393 - 0.919i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.995 + 0.0896i)T \)
47 \( 1 + (0.134 - 0.990i)T \)
53 \( 1 + (-0.963 - 0.266i)T \)
59 \( 1 + (0.936 - 0.351i)T \)
61 \( 1 + (-0.393 + 0.919i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.473 + 0.880i)T \)
73 \( 1 + (-0.550 - 0.834i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.393 + 0.919i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.473 - 0.880i)T \)
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

−24.421359149686958474776716726763, −23.63249998190406267460378582948, −22.55414588993054493425206432907, −21.82676777790909778405286269726, −20.62140225593448483216667318295, −19.83327898220409183456917018085, −19.051594653777762768016689834, −18.357601131361234352836044283238, −17.23243573298230434258472415466, −16.10519118033418763827439390447, −15.639373525403291804882657743440, −14.898674789154972361292012440117, −14.31560922481979358372999166645, −13.027186473047796857120563124399, −12.13022374963540292143160331661, −10.68490202935860220677258311933, −9.919716720836773141837626697713, −8.720577785295923899874915881624, −8.232684870661098207764981854427, −7.69265897783619419159203474174, −6.202965916674687310648724580490, −5.13052108526323621298208402027, −4.25093157523133964681292646365, −3.15531049648865673570344414071, −1.59065297229512258210702413499, 0.592990951769304936383144424326, 1.88008631176848648895363549793, 2.99500020707215967522037864448, 3.912796363607101399808621405880, 4.62506530387075087598538928220, 6.95072414349872161337598152669, 7.42152850726906566574261565149, 8.347551043589057785710471682738, 9.22068450966973672617103314983, 10.17997865643828926142883239138, 11.37735254379133780116462395355, 11.81109684410754098846074042266, 12.97895577616038139995374500845, 13.845243337681465508769705855026, 14.40209084837867283961807574961, 15.7250691130494787433436917506, 16.68210603302301482909798859557, 17.88429038562169658466445088348, 18.48086066568671698647734240776, 19.462467602326435688117462995333, 19.898469395822698829289759787297, 20.55355522660501574139907951130, 21.49883507056085873848779734754, 22.54181875963937151919606664930, 23.63565195261495879517399204944

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