↑

Properties

Label 1-53-53.26-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

Related objects

Downloads

Learn more

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: $\chi_{53} (26, \cdot )$
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 \( 1 + (0.568 - 0.822i)T \)
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

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

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