Properties

Label 1-1375-1375.103-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.736 - 0.676i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.982 + 0.187i)2-s + (−0.998 + 0.0627i)3-s + (0.929 − 0.368i)4-s + (0.968 − 0.248i)6-s + (−0.587 + 0.809i)7-s + (−0.844 + 0.535i)8-s + (0.992 − 0.125i)9-s + (−0.904 + 0.425i)12-s + (−0.125 − 0.992i)13-s + (0.425 − 0.904i)14-s + (0.728 − 0.684i)16-s + (−0.998 − 0.0627i)17-s + (−0.951 + 0.309i)18-s + (−0.968 + 0.248i)19-s + (0.535 − 0.844i)21-s + ⋯
L(s)  = 1  + (−0.982 + 0.187i)2-s + (−0.998 + 0.0627i)3-s + (0.929 − 0.368i)4-s + (0.968 − 0.248i)6-s + (−0.587 + 0.809i)7-s + (−0.844 + 0.535i)8-s + (0.992 − 0.125i)9-s + (−0.904 + 0.425i)12-s + (−0.125 − 0.992i)13-s + (0.425 − 0.904i)14-s + (0.728 − 0.684i)16-s + (−0.998 − 0.0627i)17-s + (−0.951 + 0.309i)18-s + (−0.968 + 0.248i)19-s + (0.535 − 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.736 - 0.676i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1477093295 - 0.05752688639i\)
\(L(\frac12)\) \(\approx\) \(0.1477093295 - 0.05752688639i\)
\(L(1)\) \(\approx\) \(0.3757473386 + 0.08494550299i\)
\(L(1)\) \(\approx\) \(0.3757473386 + 0.08494550299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.982 + 0.187i)T \)
3 \( 1 + (-0.998 + 0.0627i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (-0.125 - 0.992i)T \)
17 \( 1 + (-0.998 - 0.0627i)T \)
19 \( 1 + (-0.968 + 0.248i)T \)
23 \( 1 + (-0.684 + 0.728i)T \)
29 \( 1 + (0.637 + 0.770i)T \)
31 \( 1 + (0.535 + 0.844i)T \)
37 \( 1 + (0.125 + 0.992i)T \)
41 \( 1 + (-0.992 + 0.125i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.248 - 0.968i)T \)
53 \( 1 + (-0.248 + 0.968i)T \)
59 \( 1 + (-0.876 - 0.481i)T \)
61 \( 1 + (0.876 - 0.481i)T \)
67 \( 1 + (-0.998 - 0.0627i)T \)
71 \( 1 + (0.0627 + 0.998i)T \)
73 \( 1 + (0.982 - 0.187i)T \)
79 \( 1 + (-0.535 + 0.844i)T \)
83 \( 1 + (-0.844 + 0.535i)T \)
89 \( 1 + (0.425 - 0.904i)T \)
97 \( 1 + (0.368 + 0.929i)T \)
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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

−20.71545120674118656652178060494, −19.69558100786318071189055432439, −19.219058777969761196318713275061, −18.4285190790212889035615513849, −17.57025403641662048502757779234, −17.09240080476340391705843463833, −16.37904928944520054081899031710, −15.877410361244496491619594270617, −14.91943861552427487174929188445, −13.622833150805235682075317757214, −12.87357227675495755841950852418, −12.0483433737230958883434893697, −11.31173380840947582884104478525, −10.64522147583086122525954574880, −9.98479076765388499072687112385, −9.244267624514435277114755301730, −8.21063342619442945566622131134, −7.257646206271152094347137015231, −6.47691494132025038661775616289, −6.20638401144331225280119186252, −4.54003428446350839892740550466, −3.99463501291224752856980054131, −2.523709726224301704832265986101, −1.622118741069803255320094511589, −0.42599546864655991508690787029, 0.11084742110900668757181417386, 1.351075974635305044115508555190, 2.36725147061461400224109125021, 3.44413848944987774831119903761, 4.89255359704918986894240751137, 5.65374046722776096526985620325, 6.44478357006246068426427666706, 6.954738069138870493384686394908, 8.16870645016341260740637144811, 8.81069128163861301538178370466, 9.87366422017925308532138025506, 10.33484616712710443894511053081, 11.1501038917744669261994234960, 12.03535647636182244008190758163, 12.51167986184592672202975525148, 13.53510411137891317259644960055, 14.99352003878490654653328338755, 15.53681077486401627879583078057, 16.00084089943371163341454546069, 16.988706570262876027536583944761, 17.4918809718051182176737558840, 18.27893330912276889983668293964, 18.75110870321249922185547527950, 19.71540480623976206680625252386, 20.30409105195138919139947066456

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