↑

Properties

Label 1-23e2-529.4-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (0.798 + 0.601i)2-s + (−0.596 − 0.802i)3-s + (0.275 + 0.961i)4-s + (0.251 − 0.967i)5-s + (0.00620 − 0.999i)6-s + (0.988 − 0.148i)7-s + (−0.358 + 0.933i)8-s + (−0.287 + 0.957i)9-s + (0.783 − 0.621i)10-s + (0.931 − 0.363i)11-s + (0.606 − 0.794i)12-s + (0.323 + 0.946i)13-s + (0.879 + 0.476i)14-s + (−0.926 + 0.375i)15-s + (−0.847 + 0.530i)16-s + (0.890 − 0.454i)17-s + ⋯
L(s)  = 1  + (0.798 + 0.601i)2-s + (−0.596 − 0.802i)3-s + (0.275 + 0.961i)4-s + (0.251 − 0.967i)5-s + (0.00620 − 0.999i)6-s + (0.988 − 0.148i)7-s + (−0.358 + 0.933i)8-s + (−0.287 + 0.957i)9-s + (0.783 − 0.621i)10-s + (0.931 − 0.363i)11-s + (0.606 − 0.794i)12-s + (0.323 + 0.946i)13-s + (0.879 + 0.476i)14-s + (−0.926 + 0.375i)15-s + (−0.847 + 0.530i)16-s + (0.890 − 0.454i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.144274482 + 0.01273443508i\)
\(L(\frac12)\) \(\approx\) \(2.144274482 + 0.01273443508i\)
\(L(1)\) \(\approx\) \(1.604872328 + 0.06614348892i\)
\(L(1)\) \(\approx\) \(1.604872328 + 0.06614348892i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.798 + 0.601i)T \)
3 \( 1 + (-0.596 - 0.802i)T \)
5 \( 1 + (0.251 - 0.967i)T \)
7 \( 1 + (0.988 - 0.148i)T \)
11 \( 1 + (0.931 - 0.363i)T \)
13 \( 1 + (0.323 + 0.946i)T \)
17 \( 1 + (0.890 - 0.454i)T \)
19 \( 1 + (-0.977 + 0.209i)T \)
29 \( 1 + (-0.885 - 0.465i)T \)
31 \( 1 + (0.437 + 0.899i)T \)
37 \( 1 + (0.645 - 0.763i)T \)
41 \( 1 + (0.998 + 0.0496i)T \)
43 \( 1 + (-0.556 + 0.831i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (0.783 + 0.621i)T \)
59 \( 1 + (0.346 - 0.938i)T \)
61 \( 1 + (-0.311 - 0.950i)T \)
67 \( 1 + (-0.492 - 0.870i)T \)
71 \( 1 + (-0.972 - 0.233i)T \)
73 \( 1 + (-0.885 + 0.465i)T \)
79 \( 1 + (0.992 - 0.123i)T \)
83 \( 1 + (-0.896 + 0.443i)T \)
89 \( 1 + (0.969 + 0.245i)T \)
97 \( 1 + (-0.616 + 0.787i)T \)
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

−23.226666751885205589658352241208, −22.39809983310133527507648344825, −21.97546706396198455952919839926, −21.06570204098720940478741898037, −20.549479670424632606264032768209, −19.413523744715741534697732855595, −18.41426713867306370520103822573, −17.622938245511990310365183148303, −16.75823976558102526058671664150, −15.29773996922356205733134017351, −14.91199169799046546860100711679, −14.37120882271533870799256310369, −13.1334053215671677024490770026, −11.99210060879779416176042142893, −11.37894068725773008737219642427, −10.57369010433212244381606145492, −10.04792191049434380675588711611, −8.87951995363105062863216687743, −7.33317265643872180938241848970, −6.10440242919414206255065404097, −5.62752228787594069286548454571, −4.414825610081258833849549713296, −3.70301866077322883664402657915, −2.57436600528658065789541059642, −1.285008542303712464739793444211, 1.220121999812264826204614373713, 2.1571950013838331092490481891, 3.97827612988012687765398640357, 4.76263657164230976138046905878, 5.68512412850668688030671940721, 6.410251137186612061202597605171, 7.49350590331152864219199839643, 8.29034991201614369987761341935, 9.14125868690933347944390481479, 10.94342289200756713829861885779, 11.766165321998975504223216353401, 12.28123916144761173205264468578, 13.28163760098255904300524819714, 14.01952790842692407109765579958, 14.639753915341442399684325724891, 16.13242381579868234062430142311, 16.77951575177888742863201051293, 17.228715542512048258350577706394, 18.15249278091130304085373980212, 19.264117909336651829727487447864, 20.3440655770837965120818043045, 21.29531156329792807548385552845, 21.71568895761024939174453492018, 23.07961375837996814055902688227, 23.49060347243060358752493278821

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