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Properties

Label 1-23e2-529.386-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

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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: $\chi_{529} (386, \cdot )$
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 \( 1 + (0.783 - 0.621i)T \)
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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

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

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