↑

Properties

Label 1-525-525.416-r0-0-0
Degree $1$
Conductor $525$
Sign $0.307 + 0.951i$
Analytic cond. $2.43808$
Root an. cond. $2.43808$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.809 + 0.587i)8-s + (−0.669 + 0.743i)11-s + (−0.309 + 0.951i)13-s + (0.669 + 0.743i)16-s + (−0.104 + 0.994i)17-s + (−0.913 + 0.406i)19-s + (−0.809 + 0.587i)22-s + (0.978 + 0.207i)23-s + (−0.5 + 0.866i)26-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (0.5 + 0.866i)32-s + (−0.309 + 0.951i)34-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (0.809 + 0.587i)8-s + (−0.669 + 0.743i)11-s + (−0.309 + 0.951i)13-s + (0.669 + 0.743i)16-s + (−0.104 + 0.994i)17-s + (−0.913 + 0.406i)19-s + (−0.809 + 0.587i)22-s + (0.978 + 0.207i)23-s + (−0.5 + 0.866i)26-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (0.5 + 0.866i)32-s + (−0.309 + 0.951i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(2.43808\)
Root analytic conductor: \(2.43808\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.944575946 + 1.414432074i\)
\(L(\frac12)\) \(\approx\) \(1.944575946 + 1.414432074i\)
\(L(1)\) \(\approx\) \(1.723655777 + 0.5846666912i\)
\(L(1)\) \(\approx\) \(1.723655777 + 0.5846666912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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

−23.2468112771448994027349245404, −22.578926638789145426412285913991, −21.58392160564949406965006629284, −21.06183931856425530359646064981, −20.08902584264560306413100745711, −19.383583863010386078850177815882, −18.411697163684378569394781762211, −17.3471995741539922731127521942, −16.199385938810662191272221698101, −15.64012581141831030268098181439, −14.66648090953554528103724268064, −13.88285823394623451539815850280, −12.933855837624285251292877059035, −12.42234898059858171097685135676, −11.064403510761533265491123875231, −10.748552746281975377181409334435, −9.50711404773883562116296971565, −8.23327680014780463801934950534, −7.229578201123308296937619176094, −6.24635810404991426343495876964, −5.24716204085657496827784627888, −4.545524657609013925806975642757, −3.10113788151339570203687534393, −2.61315821352101602853832010471, −0.93788862502662770995780284711, 1.796475176616267240469559366713, 2.65285929540430544350956809473, 4.03227988711907109779737986070, 4.65377992462452131596987286467, 5.80806168424558136060081167351, 6.68545866578201925885843243328, 7.58331837133550865620417428262, 8.55470189540759146106355526384, 9.91713892444894012328148074623, 10.829713582548744963409737824644, 11.77359562295684523571518959358, 12.67198621533926400244630897635, 13.27729612753384156026036435312, 14.36800085221690587799781530200, 15.05626091371582358435455765987, 15.77532484967133956336473095942, 16.89516109152887006534600596199, 17.39040984371705743504307929590, 18.83554247819372818516566205631, 19.54434411889819011780836560538, 20.667725702180602226301761169961, 21.21587941969358766626635976395, 21.99648430775671469482707783458, 22.98486388622059912780996381713, 23.5897144174401943889922222914

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