↑

Properties

Label 1-1375-1375.1071-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.676 + 0.736i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.876 − 0.481i)2-s + (−0.637 − 0.770i)3-s + (0.535 − 0.844i)4-s + (−0.929 − 0.368i)6-s + (−0.809 − 0.587i)7-s + (0.0627 − 0.998i)8-s + (−0.187 + 0.982i)9-s + (−0.992 + 0.125i)12-s + (−0.187 + 0.982i)13-s + (−0.992 − 0.125i)14-s + (−0.425 − 0.904i)16-s + (−0.637 + 0.770i)17-s + (0.309 + 0.951i)18-s + (−0.929 − 0.368i)19-s + (0.0627 + 0.998i)21-s + ⋯
L(s)  = 1  + (0.876 − 0.481i)2-s + (−0.637 − 0.770i)3-s + (0.535 − 0.844i)4-s + (−0.929 − 0.368i)6-s + (−0.809 − 0.587i)7-s + (0.0627 − 0.998i)8-s + (−0.187 + 0.982i)9-s + (−0.992 + 0.125i)12-s + (−0.187 + 0.982i)13-s + (−0.992 − 0.125i)14-s + (−0.425 − 0.904i)16-s + (−0.637 + 0.770i)17-s + (0.309 + 0.951i)18-s + (−0.929 − 0.368i)19-s + (0.0627 + 0.998i)21-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5038638679 + 0.2214018340i\)
\(L(\frac12)\) \(\approx\) \(0.5038638679 + 0.2214018340i\)
\(L(1)\) \(\approx\) \(0.8894652616 - 0.4636752943i\)
\(L(1)\) \(\approx\) \(0.8894652616 - 0.4636752943i\)

Euler product

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

−20.97669780719768498911249289023, −20.266227027765608586578230393585, −19.42529064043849111787487525407, −18.1358289255595112538428174870, −17.57369036349098883681942892392, −16.65707296830614678665789293746, −16.00696657233101891430606154734, −15.53796413317886901619637985294, −14.82901544700167956901457753098, −13.99854533928142717733142918862, −12.88771124190410744105048203286, −12.40306799821891308635902119401, −11.7113410849968497332467332580, −10.69206216987548527968713904350, −10.07990451249152406415391214453, −8.92806731030165391001289358555, −8.26909283973855566933631244488, −6.872923860533304170312669470012, −6.40369894887006892173907848450, −5.51081811492407783107189755834, −4.91103157694978540536796439727, −3.959755097750017151989691262426, −3.14437602163544598129337276810, −2.3002055056431265199010153345, −0.163055936904293778485467842464, 1.27727258868588304697427298314, 2.08287887553179183375065583999, 3.10684050639662490353270534435, 4.22040847310679303435878975524, 4.813158061802649879426119514395, 6.11457437853139413800708218069, 6.45889141271154761481197981688, 7.15323899736600172617337947944, 8.29529968903142905198996489974, 9.60402227847475662006043314156, 10.30239602604745443708435331145, 11.20294776129826487975120398912, 11.70777015656173101311193924464, 12.65099559792125434576192038567, 13.20944274366397502929287155753, 13.71314818241342683908015419568, 14.64194646399023615267271188229, 15.60244056911723692402885812393, 16.4279858327147254973730296927, 17.06891094653728315446645272942, 17.987375896175743271015161706298, 19.05705293662968512797810516313, 19.41565284273049880134682380017, 19.99420510005242417624585309443, 21.12361683550709037276619360859

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