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Properties

Label 1-1375-1375.398-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.950 + 0.312i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.844 + 0.535i)2-s + (0.982 − 0.187i)3-s + (0.425 − 0.904i)4-s + (−0.728 + 0.684i)6-s + (0.951 + 0.309i)7-s + (0.125 + 0.992i)8-s + (0.929 − 0.368i)9-s + (0.248 − 0.968i)12-s + (0.368 + 0.929i)13-s + (−0.968 + 0.248i)14-s + (−0.637 − 0.770i)16-s + (−0.982 − 0.187i)17-s + (−0.587 + 0.809i)18-s + (0.728 − 0.684i)19-s + (0.992 + 0.125i)21-s + ⋯
L(s)  = 1  + (−0.844 + 0.535i)2-s + (0.982 − 0.187i)3-s + (0.425 − 0.904i)4-s + (−0.728 + 0.684i)6-s + (0.951 + 0.309i)7-s + (0.125 + 0.992i)8-s + (0.929 − 0.368i)9-s + (0.248 − 0.968i)12-s + (0.368 + 0.929i)13-s + (−0.968 + 0.248i)14-s + (−0.637 − 0.770i)16-s + (−0.982 − 0.187i)17-s + (−0.587 + 0.809i)18-s + (0.728 − 0.684i)19-s + (0.992 + 0.125i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.950 + 0.312i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.950 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.777161610 + 0.2843420139i\)
\(L(\frac12)\) \(\approx\) \(1.777161610 + 0.2843420139i\)
\(L(1)\) \(\approx\) \(1.178785957 + 0.1718890803i\)
\(L(1)\) \(\approx\) \(1.178785957 + 0.1718890803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.844 + 0.535i)T \)
3 \( 1 + (0.982 - 0.187i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (0.368 + 0.929i)T \)
17 \( 1 + (-0.982 - 0.187i)T \)
19 \( 1 + (0.728 - 0.684i)T \)
23 \( 1 + (-0.770 - 0.637i)T \)
29 \( 1 + (0.876 - 0.481i)T \)
31 \( 1 + (-0.992 + 0.125i)T \)
37 \( 1 + (0.368 + 0.929i)T \)
41 \( 1 + (0.929 - 0.368i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (0.684 - 0.728i)T \)
53 \( 1 + (-0.684 + 0.728i)T \)
59 \( 1 + (-0.0627 - 0.998i)T \)
61 \( 1 + (-0.0627 + 0.998i)T \)
67 \( 1 + (0.982 + 0.187i)T \)
71 \( 1 + (-0.187 - 0.982i)T \)
73 \( 1 + (0.844 - 0.535i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (0.125 + 0.992i)T \)
89 \( 1 + (-0.968 + 0.248i)T \)
97 \( 1 + (0.904 + 0.425i)T \)
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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

−20.46723921717423009558445993061, −20.139031726165060654380388485544, −19.51620168893265435310985381888, −18.461014856925993346012504251718, −17.94352863847332831528253515428, −17.33438047579456483015664338708, −16.04909955737650471762972336301, −15.76030302317091089097607720850, −14.642485660931999099807629017438, −13.97049862057446506190972065612, −13.05059855014833017668577305211, −12.37963837391815653937665488104, −11.192510619621115839575127997032, −10.73562122103258838395183401421, −9.84579230480211502061947902733, −9.119811461348284077101020616443, −8.25230008944719551264592629763, −7.81282159217782447483252226995, −7.08151544992292942883349746556, −5.71091838615776300252193835805, −4.39486051245638155608160086418, −3.734478331903605156166034952228, −2.77613905211002236284730800667, −1.89186565863016766628674826075, −1.07065283120878234513079401826, 1.01601282063148305811947522782, 2.012408371216120561260920201828, 2.58464873185807206501608092496, 4.131275513903320680095214314697, 4.88234027469401897907534112604, 6.09937280308617220254956017077, 6.93853847036992893666998266137, 7.65152710867201791945760082253, 8.46513317344232447650449607749, 8.984527749332866723929083253310, 9.64768648404340077995289019605, 10.738776115367193812007745456068, 11.4389883633206074576838548653, 12.35219609982385675535980329683, 13.755858818820494197800928486041, 14.008859404426491989291343134022, 14.885523063449938780837768638115, 15.60421816975396831844664121127, 16.14989720198408533613266084434, 17.27386787659405908487909569795, 18.05752195322900001808296665455, 18.46187297519090777913000139961, 19.27955033467849147010144394059, 20.08504518266123045049121485432, 20.56215231059170331351635437835

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