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Properties

Label 1-53-53.51-r1-0-0
Degree $1$
Conductor $53$
Sign $0.288 + 0.957i$
Analytic cond. $5.69564$
Root an. cond. $5.69564$
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  + (−0.992 − 0.120i)2-s + (0.464 − 0.885i)3-s + (0.970 + 0.239i)4-s + (−0.822 + 0.568i)5-s + (−0.568 + 0.822i)6-s + (−0.120 + 0.992i)7-s + (−0.935 − 0.354i)8-s + (−0.568 − 0.822i)9-s + (0.885 − 0.464i)10-s + (0.748 + 0.663i)11-s + (0.663 − 0.748i)12-s + (−0.970 + 0.239i)13-s + (0.239 − 0.970i)14-s + (0.120 + 0.992i)15-s + (0.885 + 0.464i)16-s + (0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (−0.992 − 0.120i)2-s + (0.464 − 0.885i)3-s + (0.970 + 0.239i)4-s + (−0.822 + 0.568i)5-s + (−0.568 + 0.822i)6-s + (−0.120 + 0.992i)7-s + (−0.935 − 0.354i)8-s + (−0.568 − 0.822i)9-s + (0.885 − 0.464i)10-s + (0.748 + 0.663i)11-s + (0.663 − 0.748i)12-s + (−0.970 + 0.239i)13-s + (0.239 − 0.970i)14-s + (0.120 + 0.992i)15-s + (0.885 + 0.464i)16-s + (0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.288 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5753880322 + 0.4276676398i\)
\(L(\frac12)\) \(\approx\) \(0.5753880322 + 0.4276676398i\)
\(L(1)\) \(\approx\) \(0.6589116885 + 0.05921154981i\)
\(L(1)\) \(\approx\) \(0.6589116885 + 0.05921154981i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (-0.992 - 0.120i)T \)
3 \( 1 + (0.464 - 0.885i)T \)
5 \( 1 + (-0.822 + 0.568i)T \)
7 \( 1 + (-0.120 + 0.992i)T \)
11 \( 1 + (0.748 + 0.663i)T \)
13 \( 1 + (-0.970 + 0.239i)T \)
17 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 + (0.239 + 0.970i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.748 - 0.663i)T \)
31 \( 1 + (0.663 + 0.748i)T \)
37 \( 1 + (-0.885 - 0.464i)T \)
41 \( 1 + (-0.663 + 0.748i)T \)
43 \( 1 + (-0.885 + 0.464i)T \)
47 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (-0.568 + 0.822i)T \)
61 \( 1 + (-0.935 - 0.354i)T \)
67 \( 1 + (0.239 - 0.970i)T \)
71 \( 1 + (0.464 + 0.885i)T \)
73 \( 1 + (-0.935 + 0.354i)T \)
79 \( 1 + (0.992 - 0.120i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.568 + 0.822i)T \)
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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

−32.77414570844346828958554975494, −32.09670160624178035758896155195, −30.49097185364895310893208687413, −29.20590841550024623142785114158, −27.84531326418883483716494309265, −27.06312981520333603677615266807, −26.48114439965760985406781644650, −25.00948776347446527520159422465, −23.97952724838867942607294176532, −22.356812209412618991178684911349, −20.69001294942127341665385757055, −19.97706696857304618264713580288, −19.17057842794151625459817063428, −17.12479539359213964217622771837, −16.44241421791780234316834754562, −15.389026710696739515570039944345, −14.045062645577270086702293323443, −11.88514540090278053752108900666, −10.66742144830050768856486961216, −9.44603500111729668505606524529, −8.34442850053801190626746235002, −7.1066136945643281653394452017, −4.79046098767515199860177189628, −3.1556477721207900860252702280, −0.525896374663741229533957269730, 1.81154373332908155647940556228, 3.26561259271257117444660037120, 6.348917438888148654524244589366, 7.48330841265418801051539536364, 8.53231579731187979049481773688, 9.903082132273672599052075537200, 11.87398403091445698914224116731, 12.228782076067739575046594094325, 14.57964663391645517600112401804, 15.42822702342720504612025513134, 17.17496100691801734622449534965, 18.33854968881601739806579125028, 19.2880631300815623506139122207, 19.82909012283710876706836740287, 21.53426305337381814187235612066, 23.14662747293997964323642162175, 24.56813853342380437970556939544, 25.32895349939448382376331170006, 26.45621745363866236410349501042, 27.58063332955481717729112740040, 28.720929038624003661947027111124, 29.941242876304261586903736960552, 30.78507883940699238012938787909, 31.81801798433159974947282711504, 33.82069737403805776486779989740

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