Properties

Label 1-23e2-529.380-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

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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: $\chi_{529} (380, \cdot )$
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 \( 1 + (0.311 - 0.950i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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