Properties

Label 1-1375-1375.29-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.106 + 0.994i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.425 + 0.904i)2-s + (0.929 − 0.368i)3-s + (−0.637 − 0.770i)4-s + (−0.0627 + 0.998i)6-s + (−0.809 − 0.587i)7-s + (0.968 − 0.248i)8-s + (0.728 − 0.684i)9-s + (−0.876 − 0.481i)12-s + (0.728 − 0.684i)13-s + (0.876 − 0.481i)14-s + (−0.187 + 0.982i)16-s + (−0.929 − 0.368i)17-s + (0.309 + 0.951i)18-s + (−0.0627 + 0.998i)19-s + (−0.968 − 0.248i)21-s + ⋯
L(s)  = 1  + (−0.425 + 0.904i)2-s + (0.929 − 0.368i)3-s + (−0.637 − 0.770i)4-s + (−0.0627 + 0.998i)6-s + (−0.809 − 0.587i)7-s + (0.968 − 0.248i)8-s + (0.728 − 0.684i)9-s + (−0.876 − 0.481i)12-s + (0.728 − 0.684i)13-s + (0.876 − 0.481i)14-s + (−0.187 + 0.982i)16-s + (−0.929 − 0.368i)17-s + (0.309 + 0.951i)18-s + (−0.0627 + 0.998i)19-s + (−0.968 − 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.106 + 0.994i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9559885348 + 1.063523647i\)
\(L(\frac12)\) \(\approx\) \(0.9559885348 + 1.063523647i\)
\(L(1)\) \(\approx\) \(0.9571436532 + 0.2240036917i\)
\(L(1)\) \(\approx\) \(0.9571436532 + 0.2240036917i\)

Euler product

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

−20.44444686575571368486835507837, −19.643944757824191821446429089554, −19.04313001443653479562114817881, −18.60936452824582176264599682296, −17.58286642943029428911567926554, −16.676879301750376406212078698579, −15.80420339555183816471091166119, −15.291463067434259101952400863249, −14.07447120364854931356936377656, −13.441653937823738770957897411071, −12.85325344111689194626647935982, −11.95577227830592867438444708763, −10.99458467760834967703303659135, −10.32443633275414877990788780565, −9.41143801610518451773109855866, −8.86388026086908140600080890753, −8.41357508360692553604948689490, −7.208768891828927902362112569164, −6.33444783488128181274121764705, −4.85343497582086040547575480603, −4.10649521521270381491146451455, −3.276434469319161781549917794686, −2.47286207841711294238932342052, −1.790468146795492387689278183202, −0.32976444724805260751086391828, 0.87194006563952076336908100234, 1.75777014099599893868215556782, 3.2101198785899640866044651775, 3.835464209092397592634689542971, 4.98414846114558077357972720174, 6.1493014679788666687372690907, 6.76715283836501632016672278070, 7.50839123020027972049266928054, 8.313592234356135308254849597549, 8.9257115502234239653964180765, 9.89498578591397351833090557393, 10.2941803230842882228241770520, 11.57621390389795502209587211708, 13.02145672561368199086904859247, 13.245717652472279837097378145169, 14.00444330862590695554723202656, 14.862142691192927278727572122, 15.5947129769883206086395763497, 16.12837456854994146643124156090, 17.08033189845781862965244898622, 17.87500553175565828381974641836, 18.61792150053650086344680714896, 19.18476673330330351208094722452, 20.08912591471170591459595640408, 20.39773002769405040255023711331

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