Properties

Label 2-525-21.20-c1-0-29
Degree $2$
Conductor $525$
Sign $0.544 + 0.839i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.09i·2-s + (1.70 − 0.323i)3-s + 0.791·4-s + (−0.355 − 1.87i)6-s + (1 + 2.44i)7-s − 3.06i·8-s + (2.79 − 1.09i)9-s + 3.06i·11-s + (1.34 − 0.255i)12-s − 2.44i·13-s + (2.69 − 1.09i)14-s − 1.79·16-s − 2.69·17-s + (−1.20 − 3.06i)18-s + 4.38i·19-s + ⋯
L(s)  = 1  − 0.777i·2-s + (0.982 − 0.186i)3-s + 0.395·4-s + (−0.144 − 0.763i)6-s + (0.377 + 0.925i)7-s − 1.08i·8-s + (0.930 − 0.366i)9-s + 0.925i·11-s + (0.388 − 0.0737i)12-s − 0.679i·13-s + (0.719 − 0.293i)14-s − 0.447·16-s − 0.653·17-s + (−0.284 − 0.723i)18-s + 1.00i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.544 + 0.839i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.544 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07785 - 1.12920i\)
\(L(\frac12)\) \(\approx\) \(2.07785 - 1.12920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.323i)T \)
5 \( 1 \)
7 \( 1 + (-1 - 2.44i)T \)
good2 \( 1 + 1.09iT - 2T^{2} \)
11 \( 1 - 3.06iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 - 4.38iT - 19T^{2} \)
23 \( 1 + 5.26iT - 23T^{2} \)
29 \( 1 + 5.26iT - 29T^{2} \)
31 \( 1 - 6.83iT - 31T^{2} \)
37 \( 1 + 8.58T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 + 2.69T + 47T^{2} \)
53 \( 1 - 3.93iT - 53T^{2} \)
59 \( 1 - 7.51T + 59T^{2} \)
61 \( 1 + 6.83iT - 61T^{2} \)
67 \( 1 + 4.16T + 67T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + 16.1iT - 73T^{2} \)
79 \( 1 - 0.582T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 7.51T + 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52705596669300936653233827567, −10.01952471990102179938953070907, −8.970001132797234004070895844727, −8.193691995956905966784282171187, −7.20794963473525024753526443403, −6.27046746319538881498878689837, −4.78048416539688629214989722120, −3.54205074566745684303065796964, −2.48218510391075962782045019821, −1.70946924004765435134188156464, 1.78151578642576374639183407985, 3.15803913504395878615806680332, 4.31508523882146213789451170327, 5.44578560018060438042243814153, 6.87888688228404847939200626128, 7.22228281580004200523058795502, 8.322742834739307581618142809863, 8.871443233604009003909222700206, 10.03597942986856504058147893817, 11.05341415430166242175512453242

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