Properties

Label 1-23e2-529.117-r0-0-0
Degree $1$
Conductor $529$
Sign $0.0342 - 0.999i$
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  + (−0.0186 − 0.999i)2-s + (0.992 + 0.123i)3-s + (−0.999 + 0.0372i)4-s + (−0.926 + 0.375i)5-s + (0.105 − 0.994i)6-s + (−0.820 − 0.571i)7-s + (0.0558 + 0.998i)8-s + (0.969 + 0.245i)9-s + (0.392 + 0.919i)10-s + (0.998 − 0.0496i)11-s + (−0.996 − 0.0868i)12-s + (−0.635 + 0.771i)13-s + (−0.556 + 0.831i)14-s + (−0.966 + 0.257i)15-s + (0.997 − 0.0744i)16-s + (−0.166 − 0.985i)17-s + ⋯
L(s)  = 1  + (−0.0186 − 0.999i)2-s + (0.992 + 0.123i)3-s + (−0.999 + 0.0372i)4-s + (−0.926 + 0.375i)5-s + (0.105 − 0.994i)6-s + (−0.820 − 0.571i)7-s + (0.0558 + 0.998i)8-s + (0.969 + 0.245i)9-s + (0.392 + 0.919i)10-s + (0.998 − 0.0496i)11-s + (−0.996 − 0.0868i)12-s + (−0.635 + 0.771i)13-s + (−0.556 + 0.831i)14-s + (−0.966 + 0.257i)15-s + (0.997 − 0.0744i)16-s + (−0.166 − 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.0342 - 0.999i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.0342 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9822573983 - 0.9491472552i\)
\(L(\frac12)\) \(\approx\) \(0.9822573983 - 0.9491472552i\)
\(L(1)\) \(\approx\) \(0.9923759344 - 0.5212010222i\)
\(L(1)\) \(\approx\) \(0.9923759344 - 0.5212010222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.0186 + 0.999i)T \)
3 \( 1 + (-0.992 - 0.123i)T \)
5 \( 1 + (0.926 - 0.375i)T \)
7 \( 1 + (0.820 + 0.571i)T \)
11 \( 1 + (-0.998 + 0.0496i)T \)
13 \( 1 + (0.635 - 0.771i)T \)
17 \( 1 + (0.166 + 0.985i)T \)
19 \( 1 + (-0.901 + 0.432i)T \)
29 \( 1 + (-0.369 + 0.929i)T \)
31 \( 1 + (-0.988 - 0.148i)T \)
37 \( 1 + (0.596 + 0.802i)T \)
41 \( 1 + (-0.664 - 0.747i)T \)
43 \( 1 + (-0.566 + 0.824i)T \)
47 \( 1 + (0.990 + 0.136i)T \)
53 \( 1 + (-0.392 + 0.919i)T \)
59 \( 1 + (0.263 + 0.964i)T \)
61 \( 1 + (-0.783 + 0.621i)T \)
67 \( 1 + (0.616 - 0.787i)T \)
71 \( 1 + (-0.645 - 0.763i)T \)
73 \( 1 + (-0.369 - 0.929i)T \)
79 \( 1 + (0.514 + 0.857i)T \)
83 \( 1 + (0.0434 - 0.999i)T \)
89 \( 1 + (0.471 + 0.882i)T \)
97 \( 1 + (-0.955 - 0.293i)T \)
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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

−24.1287168611805744690048233898, −22.796284094583299383005284366170, −22.358613898552606652436721853860, −21.25478293153394660033548557357, −19.89239828058034892304563938998, −19.52942306840012846133330288895, −18.7807501118332246838360605461, −17.7384052422929471460460446904, −16.65663261155570017336847810850, −15.86125520770678750437615841625, −15.19180508726060662389765065058, −14.59833410990215844362941940436, −13.55745983849548175350039107982, −12.56928130165008688483396251151, −12.14634688846715019674349537791, −10.24449986708200453577559024327, −9.340623929860826179745407567, −8.65631136714634851630557315971, −7.88137664878244671434492973070, −7.05143396316824790355506328803, −6.0923130275718593149407524393, −4.792253752107219438131233115987, −3.79015585143799066928735440998, −3.04989940708853425807621821411, −1.15657974925221627197318457339, 0.818755788942825180109597472915, 2.37255029640200535509561387992, 3.234930506851742184585425445903, 3.9787684409104737183792017232, 4.71781837718642329394336462084, 6.74875298660938283097883911212, 7.47250928016171483318449800137, 8.60077567823360768788995783817, 9.51473720682197059617140144544, 9.99405541694957708835895709268, 11.29626495499995270652641807538, 11.94042869463054307947401341914, 12.93574557706355812433476264216, 14.02321967242702037916403037351, 14.28144433551607129114758921356, 15.57295007441059778398962039035, 16.37523713998856945191780019791, 17.570658246525616221592807323006, 18.76696008568067518422925752449, 19.36463034130319538673588162207, 19.7857835398985143304690324564, 20.48510375059663429722012721357, 21.51321891816974430558083456680, 22.443973407529952553942995826567, 22.89483639293317997065758787923

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