↑

Properties

Label 1-53-53.45-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

Related objects

Downloads

Learn more

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: $\chi_{53} (45, \cdot )$
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 \( 1 + (-0.970 + 0.239i)T \)
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

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

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