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Properties

Label 1-53-53.33-r1-0-0
Degree $1$
Conductor $53$
Sign $-0.998 + 0.0549i$
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  + (−0.935 + 0.354i)2-s + (−0.992 − 0.120i)3-s + (0.748 − 0.663i)4-s + (0.239 − 0.970i)5-s + (0.970 − 0.239i)6-s + (0.354 + 0.935i)7-s + (−0.464 + 0.885i)8-s + (0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.568 − 0.822i)11-s + (−0.822 + 0.568i)12-s + (−0.748 − 0.663i)13-s + (−0.663 − 0.748i)14-s + (−0.354 + 0.935i)15-s + (0.120 − 0.992i)16-s + (−0.885 + 0.464i)17-s + ⋯
L(s)  = 1  + (−0.935 + 0.354i)2-s + (−0.992 − 0.120i)3-s + (0.748 − 0.663i)4-s + (0.239 − 0.970i)5-s + (0.970 − 0.239i)6-s + (0.354 + 0.935i)7-s + (−0.464 + 0.885i)8-s + (0.970 + 0.239i)9-s + (0.120 + 0.992i)10-s + (−0.568 − 0.822i)11-s + (−0.822 + 0.568i)12-s + (−0.748 − 0.663i)13-s + (−0.663 − 0.748i)14-s + (−0.354 + 0.935i)15-s + (0.120 − 0.992i)16-s + (−0.885 + 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.998 + 0.0549i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ -0.998 + 0.0549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0002169571419 + 0.007889822881i\)
\(L(\frac12)\) \(\approx\) \(0.0002169571419 + 0.007889822881i\)
\(L(1)\) \(\approx\) \(0.4139626014 + 0.002018400283i\)
\(L(1)\) \(\approx\) \(0.4139626014 + 0.002018400283i\)

Euler product

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

−33.88347294653790482617717922861, −33.17121573370231463998216201823, −30.85847121831454327950633573129, −29.895850668105790821434024274072, −29.092837490089568633383504876, −28.03431889763635179600946317456, −26.72329537693682463019751075615, −26.29175410274922908401943456302, −24.52637691535603520579874733670, −23.1790696254447651332046381187, −22.00150619254238646813825739254, −20.90660283258185155067268300099, −19.47453695937908243957938878141, −18.08383737430116074478247472099, −17.5328391524118652358836490100, −16.34430538891611454543310346627, −14.902009189762645559934051398264, −12.940982848983002325354949415982, −11.38800067204479106528752102338, −10.64434921462769855398199546108, −9.63149314945244754145774718185, −7.39099427759927756150649056295, −6.668399744544680846402224199107, −4.40868118714996046098407043546, −2.17596254154851927107126625563, 0.00642891546499595619174667281, 1.8066052803382547770064617946, 5.19842964221377601106618508869, 5.96613820023362797720407882713, 7.82672343599107267366623046306, 9.035968735827472612111935491498, 10.48534386811064849262730397801, 11.73277763389660566544691382676, 12.949652808153169075370410255237, 15.16022781952905937357311098525, 16.22829308278772279251350978202, 17.23099282265277175971018253934, 18.123077434123184864732497476773, 19.30633715967373441902751338242, 20.864370224226148800237457771326, 21.98812690445221439363349411464, 23.86775000937006438846595133347, 24.368659903114776583873737903408, 25.48734777873343013432917223061, 27.26439665332154754365345865066, 27.84781742985351543387598648712, 28.954938596815862803163553343420, 29.54317177919051526564639134330, 31.59779716008115017780662996519, 32.833947600315884938753581961364

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