↑

Properties

Label 1-23e2-529.71-r0-0-0
Degree $1$
Conductor $529$
Sign $0.255 - 0.966i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (−0.117 + 0.993i)2-s + (−0.966 − 0.257i)3-s + (−0.972 − 0.233i)4-s + (0.179 + 0.983i)5-s + (0.369 − 0.929i)6-s + (−0.944 − 0.329i)7-s + (0.346 − 0.938i)8-s + (0.867 + 0.498i)9-s + (−0.998 + 0.0620i)10-s + (−0.743 + 0.668i)11-s + (0.879 + 0.476i)12-s + (0.940 + 0.340i)13-s + (0.437 − 0.899i)14-s + (0.0806 − 0.996i)15-s + (0.890 + 0.454i)16-s + (−0.873 + 0.487i)17-s + ⋯
L(s)  = 1  + (−0.117 + 0.993i)2-s + (−0.966 − 0.257i)3-s + (−0.972 − 0.233i)4-s + (0.179 + 0.983i)5-s + (0.369 − 0.929i)6-s + (−0.944 − 0.329i)7-s + (0.346 − 0.938i)8-s + (0.867 + 0.498i)9-s + (−0.998 + 0.0620i)10-s + (−0.743 + 0.668i)11-s + (0.879 + 0.476i)12-s + (0.940 + 0.340i)13-s + (0.437 − 0.899i)14-s + (0.0806 − 0.996i)15-s + (0.890 + 0.454i)16-s + (−0.873 + 0.487i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.255 - 0.966i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.255 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002683443682 + 0.002065319707i\)
\(L(\frac12)\) \(\approx\) \(0.002683443682 + 0.002065319707i\)
\(L(1)\) \(\approx\) \(0.4055974465 + 0.2755347311i\)
\(L(1)\) \(\approx\) \(0.4055974465 + 0.2755347311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.117 + 0.993i)T \)
3 \( 1 + (-0.966 - 0.257i)T \)
5 \( 1 + (0.179 + 0.983i)T \)
7 \( 1 + (-0.944 - 0.329i)T \)
11 \( 1 + (-0.743 + 0.668i)T \)
13 \( 1 + (0.940 + 0.340i)T \)
17 \( 1 + (-0.873 + 0.487i)T \)
19 \( 1 + (-0.952 + 0.305i)T \)
29 \( 1 + (0.299 + 0.954i)T \)
31 \( 1 + (0.586 - 0.809i)T \)
37 \( 1 + (-0.926 - 0.375i)T \)
41 \( 1 + (-0.993 + 0.111i)T \)
43 \( 1 + (0.988 - 0.148i)T \)
47 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (-0.998 - 0.0620i)T \)
59 \( 1 + (0.392 + 0.919i)T \)
61 \( 1 + (-0.449 - 0.893i)T \)
67 \( 1 + (-0.0186 - 0.999i)T \)
71 \( 1 + (0.251 - 0.967i)T \)
73 \( 1 + (0.299 - 0.954i)T \)
79 \( 1 + (0.275 - 0.961i)T \)
83 \( 1 + (0.969 + 0.245i)T \)
89 \( 1 + (-0.847 + 0.530i)T \)
97 \( 1 + (-0.311 - 0.950i)T \)
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

−23.400003881042559212662931031498, −22.704311700989797974996041699111, −21.859550854582865447655094422884, −21.12514308360751143688409930300, −20.556734628925866771533030813407, −19.398545246750817042739230860067, −18.69882581941634361428491366830, −17.725664660801788850308236938599, −17.10629267889123410913718498827, −15.98838185961298268612656011915, −15.65054811741343811146771238032, −13.68172522368788911892989966550, −13.07179078507439990341809840653, −12.47811328886312552165776706575, −11.527654565748186327675824477147, −10.70185732822854891713445415125, −9.90743775546277996029132694688, −8.980043851965833231234312594830, −8.268772439059565483964633074545, −6.50086302082355479506285747455, −5.604956987634575890575107856538, −4.753785892497464533206162781573, −3.78766710209377269495097664245, −2.56742030037377180912228742954, −1.10009716323303352656396931933, 0.00247028176321163684044385737, 1.88282133929773870834731802438, 3.59538033879510186834332087911, 4.59754136410481811620708623553, 5.84385795177477162431221107371, 6.53334875409094385855406227684, 6.9789999363823039879473562374, 8.06295414048170219083725131426, 9.37948850673659876761392650590, 10.453073183013078151881417345690, 10.745995361408833465019622002756, 12.33055340918753477640103711997, 13.19781254142531763644689192555, 13.77517061326503802075843928858, 15.12743559354804140749485996050, 15.67882569293899534249323724560, 16.55158253235977047879150319739, 17.39285443285050260033630171339, 18.10908279035836678341843778132, 18.767341830749877837711491953735, 19.485662481214814779746084487405, 21.1187243060597453090400821982, 22.12537008020544925190212866213, 22.66474931732496908799072096056, 23.38184388798378407256053190658

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