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Properties

Label 1-53-53.27-r1-0-0
Degree $1$
Conductor $53$
Sign $0.424 - 0.905i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.424 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.273642415 - 1.444983196i\)
\(L(\frac12)\) \(\approx\) \(2.273642415 - 1.444983196i\)
\(L(1)\) \(\approx\) \(1.744784866 - 0.6800865884i\)
\(L(1)\) \(\approx\) \(1.744784866 - 0.6800865884i\)

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

−32.93094261298153439109740494640, −32.42430102674893962584354650773, −31.36322610225610982553754342992, −29.88867929098955918348396908875, −28.65110825693182048804703883288, −28.05274179234319567283690106459, −26.15848406123490366775480743305, −25.10545184542004507742798089553, −24.07755188798415764051671565165, −22.51912813270194219021937688383, −21.81983465676161011862097393798, −21.03041629059311618733195272612, −19.73567929290049068440004205512, −17.47899441596220575422852068990, −16.645098503339225014041008313427, −15.29593864477589368941403314577, −14.44645477190821857579543212981, −12.68125496704091336885282245624, −11.86507064275798516504948066111, −10.205491745897545821009783366257, −8.93032313536763677542976560801, −6.49533785314887528518663681782, −5.392606951910709284831258175238, −4.32951096229249571654995964904, −2.33187719341321653950412078325, 1.44486624264923790876760652599, 3.17238068552632454149289822039, 5.24989689621919371049921652137, 6.50636398261046099684457477238, 7.41417861644956861055162039374, 10.10863212043109714393437397859, 11.28498482008858623483874101042, 12.54800484240131754369839358421, 13.8405892162364987514422288613, 14.31567447657371822662766417564, 16.470658760937076574279979907277, 17.42321736124740537011519488628, 18.99896601547153176192006582931, 20.08173620482719321210968190200, 21.64143215679003541850827116792, 22.588902522519941445425621692799, 23.49106024664882745093038610887, 24.719866268901519406255570955768, 25.485261380037205922849683142, 27.22963280657516198569016891259, 29.27205868112260640686248199021, 29.4649113679003732499651686543, 30.29646533332378007069061885834, 31.67048060114815225173745083263, 32.98279590892902220189547725399

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