↑

Properties

Label 1-1375-1375.1131-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.129 + 0.991i$
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  + (−0.425 − 0.904i)2-s + (−0.637 + 0.770i)3-s + (−0.637 + 0.770i)4-s + (0.968 + 0.248i)6-s + 7-s + (0.968 + 0.248i)8-s + (−0.187 − 0.982i)9-s + (−0.187 − 0.982i)12-s + (−0.187 − 0.982i)13-s + (−0.425 − 0.904i)14-s + (−0.187 − 0.982i)16-s + (0.0627 + 0.998i)17-s + (−0.809 + 0.587i)18-s + (0.0627 + 0.998i)19-s + (−0.637 + 0.770i)21-s + ⋯
L(s)  = 1  + (−0.425 − 0.904i)2-s + (−0.637 + 0.770i)3-s + (−0.637 + 0.770i)4-s + (0.968 + 0.248i)6-s + 7-s + (0.968 + 0.248i)8-s + (−0.187 − 0.982i)9-s + (−0.187 − 0.982i)12-s + (−0.187 − 0.982i)13-s + (−0.425 − 0.904i)14-s + (−0.187 − 0.982i)16-s + (0.0627 + 0.998i)17-s + (−0.809 + 0.587i)18-s + (0.0627 + 0.998i)19-s + (−0.637 + 0.770i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.129 + 0.991i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.129 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4749118765 + 0.4167933478i\)
\(L(\frac12)\) \(\approx\) \(0.4749118765 + 0.4167933478i\)
\(L(1)\) \(\approx\) \(0.6601906063 + 0.01016224676i\)
\(L(1)\) \(\approx\) \(0.6601906063 + 0.01016224676i\)

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 \( 1 + (-0.637 + 0.770i)T \)
7 \( 1 + T \)
13 \( 1 + (-0.187 - 0.982i)T \)
17 \( 1 + (0.0627 + 0.998i)T \)
19 \( 1 + (0.0627 + 0.998i)T \)
23 \( 1 + (-0.992 + 0.125i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (-0.929 + 0.368i)T \)
37 \( 1 + (-0.992 - 0.125i)T \)
41 \( 1 + (-0.425 + 0.904i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.929 - 0.368i)T \)
53 \( 1 + (0.968 - 0.248i)T \)
59 \( 1 + (0.876 - 0.481i)T \)
61 \( 1 + (0.728 - 0.684i)T \)
67 \( 1 + (0.968 + 0.248i)T \)
71 \( 1 + (0.535 + 0.844i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (0.535 - 0.844i)T \)
89 \( 1 + (-0.187 + 0.982i)T \)
97 \( 1 + (-0.929 - 0.368i)T \)
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

−20.54785248489222374976886785858, −19.57186685141216803111632275003, −18.93957113287512132475849141110, −18.00409372923200778885013581609, −17.86254663595542378694467717574, −16.90833902949538915835676946399, −16.336672509938491845839432033273, −15.48894616720078532886192734241, −14.49596064246058643302021989968, −13.85927191255959412916448309305, −13.31473630640904946240895758058, −11.99911005125782623465854477223, −11.49499211835805531395079521492, −10.631688643393213318451185270407, −9.64966523871112971173838143886, −8.70157970514526195422874769667, −7.98137035141272329871316539359, −7.15487262776172274374415219981, −6.68683031308458850062075187896, −5.607830133569815146312902015549, −4.97440426940971986249907260656, −4.21567051194354080716287860862, −2.31162155897903430116204057882, −1.52674707768981709005355308538, −0.34603646983965786349884581376, 1.15132524772691092861444254198, 2.048510486654482222733828909272, 3.44351536436264671254327354359, 3.92071801517892014799742806111, 5.06411237877622486110855736422, 5.51481289404884831134919011280, 6.883786635708390134307249280064, 8.19100823718437663574715526452, 8.42043161708540131599473605619, 9.68574900570379336951972423825, 10.354365096208968574868977236109, 10.77632049628441980969842525702, 11.71193694427792899958001445297, 12.24916713135466590667337706743, 13.05314956741037669815549239445, 14.29833339778504970175651768275, 14.80955006169109255697313157866, 15.88991793553331338093124719591, 16.68690807916795753410222675703, 17.41540423103375114782647036731, 17.94375601278341434756986108957, 18.55663933445141571003783846214, 19.78389920584363025776995590758, 20.32223924135506083403802671196, 20.99634713888225671025550906606

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