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Properties

Label 1-1375-1375.113-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.998 + 0.0488i$
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  + (−0.770 + 0.637i)2-s + (−0.982 − 0.187i)3-s + (0.187 − 0.982i)4-s + (0.876 − 0.481i)6-s − i·7-s + (0.481 + 0.876i)8-s + (0.929 + 0.368i)9-s + (−0.368 + 0.929i)12-s + (0.368 − 0.929i)13-s + (0.637 + 0.770i)14-s + (−0.929 − 0.368i)16-s + (−0.125 + 0.992i)17-s + (−0.951 + 0.309i)18-s + (0.992 + 0.125i)19-s + (−0.187 + 0.982i)21-s + ⋯
L(s)  = 1  + (−0.770 + 0.637i)2-s + (−0.982 − 0.187i)3-s + (0.187 − 0.982i)4-s + (0.876 − 0.481i)6-s − i·7-s + (0.481 + 0.876i)8-s + (0.929 + 0.368i)9-s + (−0.368 + 0.929i)12-s + (0.368 − 0.929i)13-s + (0.637 + 0.770i)14-s + (−0.929 − 0.368i)16-s + (−0.125 + 0.992i)17-s + (−0.951 + 0.309i)18-s + (0.992 + 0.125i)19-s + (−0.187 + 0.982i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.998 + 0.0488i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.998 + 0.0488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002366371162 + 0.09675162231i\)
\(L(\frac12)\) \(\approx\) \(0.002366371162 + 0.09675162231i\)
\(L(1)\) \(\approx\) \(0.5247324539 + 0.05665315603i\)
\(L(1)\) \(\approx\) \(0.5247324539 + 0.05665315603i\)

Euler product

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

−20.6041863261290698158481166734, −19.11241510815414867000109806081, −18.79836657386660232857172371669, −18.03660311220347117585908551522, −17.47546464693277486675573540914, −16.51149352431311718916517220399, −15.98882697323625119904686340266, −15.3774512850272653769950645360, −13.99537801619050481559289753734, −13.11001533194196464409917229382, −12.05084558798368014983618066710, −11.80710255197666376292642230896, −11.15592273041029938669013315045, −10.125201800063226525125839408120, −9.50153469149533991740770839934, −8.804071088496048493193070719432, −7.798621426886261048397573071322, −6.78970862660680300353338187694, −6.13551094744913360706145634133, −4.97687336162028479618028973683, −4.23161568031709616086883951394, −3.032282205853883091048020307714, −2.111231491857504056259480888, −1.04616188090267092212107758420, −0.03633566244134768182873808669, 1.03356581565299246497670701702, 1.56351991651621480640740228677, 3.30773998025200551179566645475, 4.491052292762974321659739556874, 5.3742004545775514641721673684, 6.06901601756387246374441190285, 6.90161550807797609650951631791, 7.59852981122748308963447877266, 8.25933413769368819556062381279, 9.48887553289204812562344022418, 10.32229754670663963835969961269, 10.70941136302580168805322415698, 11.52256627110624455666972614495, 12.56687335814496834896018768450, 13.45907733797487232186530267822, 14.17118950230389696632405658894, 15.27460356109112478505678943869, 15.9538449418874761440709486458, 16.54533355406886305237492772236, 17.41837339604600507515346381, 17.7486447831310422943751373137, 18.43543777474236350916849016744, 19.52747892421295047346554236281, 19.92042106130522721707281187660, 20.944907984765102642007676743287

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