↑

Properties

Label 1-4235-4235.513-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.318 - 0.947i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.901 − 0.432i)2-s + (0.406 + 0.913i)3-s + (0.625 + 0.780i)4-s + (0.0285 − 0.999i)6-s + (−0.226 − 0.974i)8-s + (−0.669 + 0.743i)9-s + (−0.458 + 0.888i)12-s + (0.633 − 0.774i)13-s + (−0.217 + 0.976i)16-s + (−0.556 − 0.830i)17-s + (0.924 − 0.380i)18-s + (0.999 − 0.0380i)19-s + (−0.0950 − 0.995i)23-s + (0.797 − 0.603i)24-s + (−0.905 + 0.424i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.901 − 0.432i)2-s + (0.406 + 0.913i)3-s + (0.625 + 0.780i)4-s + (0.0285 − 0.999i)6-s + (−0.226 − 0.974i)8-s + (−0.669 + 0.743i)9-s + (−0.458 + 0.888i)12-s + (0.633 − 0.774i)13-s + (−0.217 + 0.976i)16-s + (−0.556 − 0.830i)17-s + (0.924 − 0.380i)18-s + (0.999 − 0.0380i)19-s + (−0.0950 − 0.995i)23-s + (0.797 − 0.603i)24-s + (−0.905 + 0.424i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.318 - 0.947i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.318 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3710124462 - 0.5159922885i\)
\(L(\frac12)\) \(\approx\) \(0.3710124462 - 0.5159922885i\)
\(L(1)\) \(\approx\) \(0.7320189880 + 0.01241017348i\)
\(L(1)\) \(\approx\) \(0.7320189880 + 0.01241017348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.901 - 0.432i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (0.633 - 0.774i)T \)
17 \( 1 + (-0.556 - 0.830i)T \)
19 \( 1 + (0.999 - 0.0380i)T \)
23 \( 1 + (-0.0950 - 0.995i)T \)
29 \( 1 + (0.696 + 0.717i)T \)
31 \( 1 + (-0.710 - 0.703i)T \)
37 \( 1 + (0.836 - 0.548i)T \)
41 \( 1 + (-0.941 - 0.336i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (-0.924 - 0.380i)T \)
53 \( 1 + (0.976 - 0.217i)T \)
59 \( 1 + (0.179 - 0.983i)T \)
61 \( 1 + (-0.997 - 0.0760i)T \)
67 \( 1 + (-0.971 - 0.235i)T \)
71 \( 1 + (-0.985 - 0.170i)T \)
73 \( 1 + (-0.318 + 0.948i)T \)
79 \( 1 + (-0.999 - 0.0190i)T \)
83 \( 1 + (-0.113 - 0.993i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.389 + 0.921i)T \)
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

−18.37426026646217370881583531183, −18.0568877394042094245081725617, −17.38846866307004291111726898991, −16.66316102300495151519736452852, −15.955758743663715403529029394584, −15.20947801523941068565459073251, −14.633977302967265044016512782406, −13.6851122246513129155895827759, −13.49978275978031390204524540204, −12.30002932145327537845624563017, −11.66117438660960456446763166898, −11.11796997907427802464884231748, −10.15964393992232100063741459299, −9.40068775564082268757520384360, −8.757688123874528103433416335137, −8.24523009681192364898600856842, −7.41880572644805610941972816489, −6.94798279531798253399159913863, −6.109589205342405933999309195444, −5.67954699271762830909112107002, −4.451170008166953758106673070380, −3.3760518482209374468637475411, −2.55254408985214764609120638792, −1.55612117930276669446861737582, −1.24386121644416841748939025404, 0.22741168098145645547301003568, 1.340667700378964404564779363599, 2.47165180738261239686425000511, 2.98094153828739717195770332239, 3.72520404354718074639910962151, 4.54892894473199053589906976364, 5.42987143145016953464780868362, 6.33914619705032387341194854687, 7.31395940899877347483689281356, 7.94906075281811301232782703204, 8.72547598285771133081479912143, 9.14802422176730932833120230090, 9.96896103972322999168075609356, 10.44607070807177185425747400314, 11.20595465420702721563258727748, 11.6531005068277093261173987016, 12.688157445840039720184484524170, 13.349952346924870111758470007418, 14.165733580637703345942685987989, 14.98583502921508468832759219094, 15.68983735556303038696626425889, 16.23853828816688888014019563333, 16.63648694837262954760557198547, 17.67286166705825937418392335100, 18.15050629680773400002653080151

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