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Properties

Label 1-53-53.12-r1-0-0
Degree $1$
Conductor $53$
Sign $-0.952 + 0.305i$
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  + (−0.663 + 0.748i)2-s + (0.239 + 0.970i)3-s + (−0.120 − 0.992i)4-s + (0.464 + 0.885i)5-s + (−0.885 − 0.464i)6-s + (0.748 + 0.663i)7-s + (0.822 + 0.568i)8-s + (−0.885 + 0.464i)9-s + (−0.970 − 0.239i)10-s + (0.354 + 0.935i)11-s + (0.935 − 0.354i)12-s + (0.120 − 0.992i)13-s + (−0.992 + 0.120i)14-s + (−0.748 + 0.663i)15-s + (−0.970 + 0.239i)16-s + (−0.568 − 0.822i)17-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (0.239 + 0.970i)3-s + (−0.120 − 0.992i)4-s + (0.464 + 0.885i)5-s + (−0.885 − 0.464i)6-s + (0.748 + 0.663i)7-s + (0.822 + 0.568i)8-s + (−0.885 + 0.464i)9-s + (−0.970 − 0.239i)10-s + (0.354 + 0.935i)11-s + (0.935 − 0.354i)12-s + (0.120 − 0.992i)13-s + (−0.992 + 0.120i)14-s + (−0.748 + 0.663i)15-s + (−0.970 + 0.239i)16-s + (−0.568 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.952 + 0.305i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ -0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1904462631 + 1.216657327i\)
\(L(\frac12)\) \(\approx\) \(0.1904462631 + 1.216657327i\)
\(L(1)\) \(\approx\) \(0.6055021319 + 0.7223880323i\)
\(L(1)\) \(\approx\) \(0.6055021319 + 0.7223880323i\)

Euler product

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

−32.2974657115140149051859174426, −31.05710877360397523759660966702, −30.10915751514542124942184188342, −29.19275223855262195463420928207, −28.30949921762540930643066581867, −26.93588331240605990220960874177, −25.80097965067269969054904374251, −24.542100468762231760592840631889, −23.69390019523311096598031747799, −21.702289931124044569495093555350, −20.6853308086533153460463177178, −19.686709685179758124545245562009, −18.62506159801703712141921987576, −17.31614633522414334947224267753, −16.71892540812299433709688421987, −14.12085337803428603794549720720, −13.17838789290814999382910597980, −12.002499079542594002903367309880, −10.76374849406372141606742733370, −8.88336893756945184023423803975, −8.24244093834743269045734824324, −6.53211749892933580193705755627, −4.217042208552584259234815837899, −2.05628450463722260138392195690, −0.88800654231865333090784449415, 2.37776698312222506273086247982, 4.73731244648094954141692134953, 6.07088594116879159351819785024, 7.75740249449866553039239983967, 9.142805633975577370800741501967, 10.18107917429820973845737895600, 11.29546816794531179179573516217, 13.881122324974319020718763898, 15.0540358838129030812123017550, 15.46499708307377986391384701181, 17.317836211073355070081093467249, 17.97201136544376550173067223280, 19.474903874454289077468095901919, 20.812236615447908467197933853, 22.1584332961351595721579335230, 23.1292330062821680039405942601, 25.07480544978015989171476376101, 25.49592774715307474510402549165, 26.83573581912641263478568425979, 27.57304981017199652716844325458, 28.56607623291653629970772704099, 30.262366071460528442251933090082, 31.62451475343641190442309104965, 32.801850634556946810773331244271, 33.83076099398303986572423833997

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