↑

Properties

Label 1-23e2-529.492-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.896 + 0.442i$
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  + (−0.907 + 0.421i)2-s + (0.275 − 0.961i)3-s + (0.645 − 0.763i)4-s + (0.0806 + 0.996i)5-s + (0.154 + 0.987i)6-s + (−0.834 − 0.551i)7-s + (−0.263 + 0.964i)8-s + (−0.847 − 0.530i)9-s + (−0.492 − 0.870i)10-s + (−0.993 + 0.111i)11-s + (−0.556 − 0.831i)12-s + (0.931 + 0.363i)13-s + (0.988 + 0.148i)14-s + (0.980 + 0.197i)15-s + (−0.166 − 0.985i)16-s + (0.717 − 0.696i)17-s + ⋯
L(s)  = 1  + (−0.907 + 0.421i)2-s + (0.275 − 0.961i)3-s + (0.645 − 0.763i)4-s + (0.0806 + 0.996i)5-s + (0.154 + 0.987i)6-s + (−0.834 − 0.551i)7-s + (−0.263 + 0.964i)8-s + (−0.847 − 0.530i)9-s + (−0.492 − 0.870i)10-s + (−0.993 + 0.111i)11-s + (−0.556 − 0.831i)12-s + (0.931 + 0.363i)13-s + (0.988 + 0.148i)14-s + (0.980 + 0.197i)15-s + (−0.166 − 0.985i)16-s + (0.717 − 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (492, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.896 + 0.442i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02327616755 + 0.09977782662i\)
\(L(\frac12)\) \(\approx\) \(0.02327616755 + 0.09977782662i\)
\(L(1)\) \(\approx\) \(0.5330721908 + 0.003527529758i\)
\(L(1)\) \(\approx\) \(0.5330721908 + 0.003527529758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.907 + 0.421i)T \)
3 \( 1 + (0.275 - 0.961i)T \)
5 \( 1 + (0.0806 + 0.996i)T \)
7 \( 1 + (-0.834 - 0.551i)T \)
11 \( 1 + (-0.993 + 0.111i)T \)
13 \( 1 + (0.931 + 0.363i)T \)
17 \( 1 + (0.717 - 0.696i)T \)
19 \( 1 + (-0.535 + 0.844i)T \)
29 \( 1 + (-0.896 - 0.443i)T \)
31 \( 1 + (-0.944 - 0.329i)T \)
37 \( 1 + (-0.966 + 0.257i)T \)
41 \( 1 + (0.323 - 0.946i)T \)
43 \( 1 + (-0.820 + 0.571i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (-0.492 + 0.870i)T \)
59 \( 1 + (0.545 - 0.837i)T \)
61 \( 1 + (-0.998 - 0.0620i)T \)
67 \( 1 + (-0.311 + 0.950i)T \)
71 \( 1 + (-0.926 - 0.375i)T \)
73 \( 1 + (-0.896 + 0.443i)T \)
79 \( 1 + (-0.999 + 0.0372i)T \)
83 \( 1 + (-0.471 + 0.882i)T \)
89 \( 1 + (0.997 + 0.0744i)T \)
97 \( 1 + (0.783 + 0.621i)T \)
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

−23.02483348685755836445448247285, −21.768892668994660563323224979929, −21.34628184209233677643093377005, −20.53885207235817043688682607438, −19.889555336424426405412214321829, −19.05231139450695499410620117851, −18.11857214951008781668950292298, −17.03330221327872841894036595375, −16.31930099965099932701117723082, −15.794230168022324962624319621816, −15.03494184705890044450352466099, −13.27813316844250055481657529003, −12.8504670924806911958800342078, −11.714351891011232187219666538618, −10.63238739875290267177100308975, −10.06035385634014059131967115863, −8.89907096497573533782291440471, −8.74274751558882328299530092353, −7.6470324271950700754735595359, −6.059223185992270023893600630645, −5.20298792563394381410679481360, −3.7954836020493139651946382885, −3.046909259464710452515011565879, −1.82313186838274567088098053974, −0.06693316128932112863982976360, 1.526885303358764492313529355140, 2.61372406901603845893660582887, 3.55339699111451457052314937680, 5.720792115486862900861998446444, 6.31236330057462015712382761046, 7.31546230341600228243285428543, 7.680273164290288948487206006725, 8.87740435543903830362618857931, 9.89845299566631643223116864535, 10.67797271225418731842921564890, 11.5440543657287811682875914819, 12.77808067737910470203184497739, 13.75955632558191162485501301373, 14.407426414452312150009601554, 15.4399989657833735646727886131, 16.31723067471745807810423867106, 17.23521424441598365939808557984, 18.19785513905786472294705545169, 18.846462211175130861247043180764, 19.06256371628843452874783496075, 20.33900278593554559117905196621, 20.90491333752733887833287029044, 22.59038987021577546229817248381, 23.309207590282659988728061538421, 23.72156251075274533007863631690

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