Properties

Label 1-53-53.16-r0-0-0
Degree $1$
Conductor $53$
Sign $0.177 + 0.984i$
Analytic cond. $0.246130$
Root an. cond. $0.246130$
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  + (0.885 + 0.464i)2-s + (−0.354 + 0.935i)3-s + (0.568 + 0.822i)4-s + (−0.748 − 0.663i)5-s + (−0.748 + 0.663i)6-s + (0.885 + 0.464i)7-s + (0.120 + 0.992i)8-s + (−0.748 − 0.663i)9-s + (−0.354 − 0.935i)10-s + (−0.970 + 0.239i)11-s + (−0.970 + 0.239i)12-s + (0.568 − 0.822i)13-s + (0.568 + 0.822i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + (−0.354 + 0.935i)3-s + (0.568 + 0.822i)4-s + (−0.748 − 0.663i)5-s + (−0.748 + 0.663i)6-s + (0.885 + 0.464i)7-s + (0.120 + 0.992i)8-s + (−0.748 − 0.663i)9-s + (−0.354 − 0.935i)10-s + (−0.970 + 0.239i)11-s + (−0.970 + 0.239i)12-s + (0.568 − 0.822i)13-s + (0.568 + 0.822i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.177 + 0.984i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.177 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8991539179 + 0.7516505647i\)
\(L(\frac12)\) \(\approx\) \(0.8991539179 + 0.7516505647i\)
\(L(1)\) \(\approx\) \(1.153237824 + 0.6315225704i\)
\(L(1)\) \(\approx\) \(1.153237824 + 0.6315225704i\)

Euler product

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

−33.298815350066214275062024924136, −31.30714563507863312634150765854, −30.939617097607794907523517266339, −29.947149866591134511738299085635, −28.92593637751900800862731326435, −27.753670874626640612572133514855, −26.204637183478932011887658517952, −24.53439103121418307842367895877, −23.55569958603501131189734641708, −23.149196349989837935170575804698, −21.627207183169158233796252160942, −20.353536097801821141112546212573, −19.04655851272990681515040182669, −18.30607689513344378894021731438, −16.45780181588924157224815367338, −14.86807220457389626805229843823, −13.8718659874542467945251968192, −12.63430159832082004924201582767, −11.32750068631229954104585519717, −10.766674921847484342655280788456, −7.970527360774459251385225052178, −6.82400371244705737355255238959, −5.31122537245342078556903535063, −3.58099007613297975173827326242, −1.74843178072362645827298977380, 3.19585609717101304775405314232, 4.82575923266381257163241235511, 5.367701981075557879110313745197, 7.59228195473776454446572124729, 8.839424771609448107476722467377, 10.99692499260237177302015922835, 11.866467958882154068762946501327, 13.27809559751477426705062474895, 15.06745430378664071264126340837, 15.57261268118008174119063941796, 16.68795329119121734155699615901, 18.01962586385899698493361070822, 20.49068053387608520822963485081, 20.7781096961209658041377905570, 22.20544771834235915444963020779, 23.25189484033139896484496342283, 24.16046678248009008967112077589, 25.46077593834938552693863420444, 26.854789794182820477542232938533, 27.83277179609271710588145965445, 28.97638056897874814444026095070, 30.74885091990031145673573882677, 31.50885645447488701849448706857, 32.45213604308256362002948753240, 33.524731969932227897581387346695

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