↑

Properties

Label 1-4235-4235.2952-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.340 - 0.940i$
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  + (0.0190 − 0.999i)2-s + (0.406 + 0.913i)3-s + (−0.999 − 0.0380i)4-s + (0.921 − 0.389i)6-s + (−0.0570 + 0.998i)8-s + (−0.669 + 0.743i)9-s + (−0.371 − 0.928i)12-s + (−0.170 − 0.985i)13-s + (0.997 + 0.0760i)16-s + (0.917 + 0.398i)17-s + (0.730 + 0.683i)18-s + (0.861 + 0.508i)19-s + (0.971 − 0.235i)23-s + (−0.935 + 0.353i)24-s + (−0.988 + 0.151i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.0190 − 0.999i)2-s + (0.406 + 0.913i)3-s + (−0.999 − 0.0380i)4-s + (0.921 − 0.389i)6-s + (−0.0570 + 0.998i)8-s + (−0.669 + 0.743i)9-s + (−0.371 − 0.928i)12-s + (−0.170 − 0.985i)13-s + (0.997 + 0.0760i)16-s + (0.917 + 0.398i)17-s + (0.730 + 0.683i)18-s + (0.861 + 0.508i)19-s + (0.971 − 0.235i)23-s + (−0.935 + 0.353i)24-s + (−0.988 + 0.151i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.340 - 0.940i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2952, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.340 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384557394 - 0.9714013735i\)
\(L(\frac12)\) \(\approx\) \(1.384557394 - 0.9714013735i\)
\(L(1)\) \(\approx\) \(1.078028140 - 0.3005634106i\)
\(L(1)\) \(\approx\) \(1.078028140 - 0.3005634106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0190 - 0.999i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (-0.170 - 0.985i)T \)
17 \( 1 + (0.917 + 0.398i)T \)
19 \( 1 + (0.861 + 0.508i)T \)
23 \( 1 + (0.971 - 0.235i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (0.0665 - 0.997i)T \)
37 \( 1 + (0.962 + 0.272i)T \)
41 \( 1 + (-0.0855 - 0.996i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (-0.730 + 0.683i)T \)
53 \( 1 + (-0.0760 - 0.997i)T \)
59 \( 1 + (0.820 + 0.572i)T \)
61 \( 1 + (-0.483 + 0.875i)T \)
67 \( 1 + (0.189 + 0.981i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (-0.983 - 0.179i)T \)
79 \( 1 + (-0.964 + 0.263i)T \)
83 \( 1 + (-0.999 - 0.0285i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (0.633 - 0.774i)T \)
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

−18.43701429542397076616837796709, −17.88602836684260958503679866272, −17.12532755042690532678785213362, −16.47264562720865701623659236332, −15.91561387078083273130060221210, −14.84128276779945815910070676379, −14.52162239931512350976144661809, −13.85999796575188856847546087211, −13.237080402242609853834178137616, −12.61012644978476122456519530829, −11.85401819128214196632150495196, −11.15562115070195307347452334082, −9.84865268803024817260221968268, −9.33310767187355644303091078678, −8.69844308284795320143719719014, −7.945598813944764367487036127634, −7.21386936563662371421722281387, −6.87940136497672126882409669335, −6.06300639373739659022656335274, −5.22676908092678573368352484752, −4.59408735777787335651030060467, −3.38998446207015128819308105219, −2.945175891296323124880642302434, −1.57275196875656422795367276055, −0.928649147455524782879773028229, 0.54629424562256220908931018952, 1.61906458620101204195839877797, 2.66262272647274145370668034673, 3.14710650795675603212203014046, 3.88038639835185525630775996288, 4.575211854360416544364567841240, 5.47795013500104133842081787038, 5.811990204972384066889505856155, 7.45472234597048131436304010224, 8.05332110054331816734302311678, 8.7219216171685782240847592407, 9.531326681407909422221166618655, 10.10909476839076754041209285755, 10.45942785952658435294704861982, 11.45039243761474225981961130251, 11.84169867313806372138878076572, 12.93084911577540550891253448059, 13.306733396237607871622645718580, 14.240601967386862680563108561507, 14.77085906977612162475002397150, 15.32513391513288534878649432793, 16.278860496673445967179856586, 16.99701102394437337550191608295, 17.5156065539334340682989955043, 18.48666853355860549456359797669

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