↑

Properties

Label 1-525-525.104-r0-0-0
Degree $1$
Conductor $525$
Sign $0.0627 + 0.998i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.309 − 0.951i)22-s + (−0.809 − 0.587i)23-s + 26-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + 32-s + (0.809 − 0.587i)34-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)8-s + (0.809 + 0.587i)11-s + (−0.809 + 0.587i)13-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (−0.309 − 0.951i)22-s + (−0.809 − 0.587i)23-s + 26-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + 32-s + (0.809 − 0.587i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.0627 + 0.998i$
Analytic conductor: \(2.43808\)
Root analytic conductor: \(2.43808\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (0:\ ),\ 0.0627 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4093658761 + 0.3844201454i\)
\(L(\frac12)\) \(\approx\) \(0.4093658761 + 0.3844201454i\)
\(L(1)\) \(\approx\) \(0.6390064304 + 0.01709394925i\)
\(L(1)\) \(\approx\) \(0.6390064304 + 0.01709394925i\)

Euler product

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

−23.639657647490463888353655769789, −22.41952638378300141049377294396, −21.85232450116254961711995949854, −20.35604686026433884257339396124, −19.88929815080704640413372281469, −19.01161405333003720354858640970, −18.12986488812803190550682274434, −17.35113332260328342605437721157, −16.63068015257771922977354182293, −15.7516277079486547237004105860, −14.92058836547166481764272641684, −14.12089097370304603540369305041, −13.15702293520795061558530492934, −11.74796402244407420174918345514, −11.132242634167624657458865346012, −9.95450184556898661864931695389, −9.28330302780091254699192299882, −8.37212181739751696275793617352, −7.39082221927526536542207990072, −6.60423575207242476189237038552, −5.57531596094578807195991136367, −4.62416339922466181460641325867, −3.066889456468444555637124101195, −1.815632142846552544872312266875, −0.38441600051669334406044921273, 1.52806740021278904313924880770, 2.29565945062757842223334439558, 3.732758330206960354113717984768, 4.464526376296443791796307752697, 6.16408108887944817421816802947, 7.04152575320582129283922899444, 8.04512606762970811945659346914, 8.91203992964403008447109357165, 9.85632810235290332855804863912, 10.47067101916254213886623460899, 11.71359133232784358194643545074, 12.20157743557781813923036087156, 13.13989808341847407621344540766, 14.39156375069930350131389067938, 15.19664291606391065206840375925, 16.523485099109506179037518035035, 16.93815704441092351837379842915, 17.87232617125315914869963208133, 18.6905800204096692181462532647, 19.64774039064669302336746045570, 20.036732672885741811837194155710, 21.17198315931879662907801074995, 21.84058884221919490509342673642, 22.62103386161813536034492297224, 23.75729099840637125888673304604

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