↑

Properties

Label 1-275-275.52-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.193 + 0.981i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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  + (−0.587 + 0.809i)2-s − i·3-s + (−0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s − 9-s + (−0.951 + 0.309i)12-s + i·13-s + (−0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.587 + 0.809i)17-s + (0.587 − 0.809i)18-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s − i·3-s + (−0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (−0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s − 9-s + (−0.951 + 0.309i)12-s + i·13-s + (−0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.587 + 0.809i)17-s + (0.587 − 0.809i)18-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.193 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3523639218 + 0.4288525979i\)
\(L(\frac12)\) \(\approx\) \(0.3523639218 + 0.4288525979i\)
\(L(1)\) \(\approx\) \(0.6125074413 + 0.1771287729i\)
\(L(1)\) \(\approx\) \(0.6125074413 + 0.1771287729i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 - iT \)
7 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + iT \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + iT \)
47 \( 1 + iT \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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

−25.82319586214025720564206696225, −24.8664936409447264972420586495, −23.06083584908998767372126396420, −22.58767762430352351994486611165, −21.711380210523367278479875499391, −20.49523706506318690804337052384, −20.2851378244065742399930217530, −19.27182883344165160326670970363, −18.027550892745428915101735074358, −17.17183813376093306252172667588, −16.32172830182385017340701817657, −15.54913637689154370859510579707, −14.073975583247288940821788612272, −13.23162023854322047202543672842, −11.98858049693380229893559915491, −11.043652158532877036482766866569, −10.063472803881962854413377409701, −9.6940275099433417043421897342, −8.40650175641548962115348980869, −7.438809979003206794502647625709, −5.812353782813821527992126676925, −4.34332258794470733889238667404, −3.58751492180018090234971172931, −2.4867474795923811130784092576, −0.471513005531091530242548902385, 1.44452596620016584387619628970, 2.672089031753781090256044480573, 4.65909414143941128375766871960, 6.08008773752796694048329712052, 6.51666990979415893149615694001, 7.63279564856494941612729260863, 8.71779967107413046947971628858, 9.324060208459904746490750165114, 10.7898198389587096907155644658, 11.920573141130617594748512499958, 12.97973910043073721607695234008, 13.93018192247269101506869264330, 14.84750393685792560469270341020, 15.92865480239783161304501380251, 16.753191780121177902134353679208, 17.9281343879418830336030940785, 18.34365551285279735811223844922, 19.52237871957619762025265682566, 19.754727274980447049771644745292, 21.63492161954003956719791010151, 22.56049252727360265857119155058, 23.63079952010359814861336930082, 24.223687963962172141142514089529, 25.01010714868217953740156331122, 25.93673994503105173189836258663

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