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Properties

Label 1-1375-1375.223-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.307 + 0.951i$
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  + (−0.998 − 0.0627i)2-s + (−0.684 + 0.728i)3-s + (0.992 + 0.125i)4-s + (0.728 − 0.684i)6-s + (0.951 − 0.309i)7-s + (−0.982 − 0.187i)8-s + (−0.0627 − 0.998i)9-s + (−0.770 + 0.637i)12-s + (−0.998 + 0.0627i)13-s + (−0.968 + 0.248i)14-s + (0.968 + 0.248i)16-s + (0.982 + 0.187i)17-s + i·18-s + (0.425 + 0.904i)19-s + (−0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (−0.998 − 0.0627i)2-s + (−0.684 + 0.728i)3-s + (0.992 + 0.125i)4-s + (0.728 − 0.684i)6-s + (0.951 − 0.309i)7-s + (−0.982 − 0.187i)8-s + (−0.0627 − 0.998i)9-s + (−0.770 + 0.637i)12-s + (−0.998 + 0.0627i)13-s + (−0.968 + 0.248i)14-s + (0.968 + 0.248i)16-s + (0.982 + 0.187i)17-s + i·18-s + (0.425 + 0.904i)19-s + (−0.425 + 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8619174303 + 0.6271148207i\)
\(L(\frac12)\) \(\approx\) \(0.8619174303 + 0.6271148207i\)
\(L(1)\) \(\approx\) \(0.6347944767 + 0.1390849381i\)
\(L(1)\) \(\approx\) \(0.6347944767 + 0.1390849381i\)

Euler product

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

−20.26764023394343355340628332779, −19.50984270540487033003154242584, −18.85485448382838181912051806713, −18.064716842125550979213595488979, −17.68558910625112560249041893741, −16.83932253847621088664528228898, −16.38065232582449737114275283485, −15.17271077464816815142203019590, −14.640494680938471937933467220429, −13.56259857763785810628667464506, −12.444534892332440703567617664502, −11.898100667711432555937333834838, −11.226326791432770441725711497739, −10.52976288912995259329262706547, −9.58751678507091873223834036356, −8.66271757250302452779524465717, −7.82044788094319479907393166267, −7.25338972752812805336852136668, −6.532282402334207463257773190227, −5.28296512105395122745809287666, −5.03252119330982494609545095125, −3.10129851829271872757581089791, −2.1785179775214540414872962236, −1.32686005421313798342091539218, −0.46134476653262881173409735421, 0.742904873468544220519230192435, 1.59536416112948170025769337852, 2.87469197986971285570010445961, 3.88985817247563396659786419789, 4.96285303904924910541230610332, 5.67949155332801761219945729523, 6.69251623502622746758709821275, 7.631476583918513377596850955652, 8.23051825270734166358128419692, 9.38708612671803593195310265146, 9.957559787183384367335487261622, 10.57248582427495539080671475776, 11.571146191909112778123183933525, 11.80104078663309855408929990912, 12.8381717255518753889914152272, 14.355967461469280625154279020877, 14.867047030712267296312190567203, 15.62563247935200834814106969892, 16.5787419735278948084576065238, 17.196652596671199321656411986633, 17.39704593449482987805200328166, 18.60152157866381648402160459744, 18.99474747244893652696849879586, 20.33521299349507671109123359801, 20.64729365169972611465525457127

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