↑

Properties

Label 2-525-105.59-c1-0-34
Degree $2$
Conductor $525$
Sign $-0.999 - 0.00342i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 1.5i)2-s + (0.866 + 1.5i)3-s + (−0.5 + 0.866i)4-s + (1.5 − 2.59i)6-s + (−0.866 − 2.5i)7-s − 1.73·8-s + (−1.5 + 2.59i)9-s + (−3 − 1.73i)11-s − 1.73·12-s − 3.46·13-s + (−3 + 3.46i)14-s + (2.49 + 4.33i)16-s + (−5.19 − 3i)17-s + 5.19·18-s + (6 − 3.46i)19-s + ⋯
L(s)  = 1  + (−0.612 − 1.06i)2-s + (0.499 + 0.866i)3-s + (−0.250 + 0.433i)4-s + (0.612 − 1.06i)6-s + (−0.327 − 0.944i)7-s − 0.612·8-s + (−0.5 + 0.866i)9-s + (−0.904 − 0.522i)11-s − 0.500·12-s − 0.960·13-s + (−0.801 + 0.925i)14-s + (0.624 + 1.08i)16-s + (−1.26 − 0.727i)17-s + 1.22·18-s + (1.37 − 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.999 - 0.00342i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.999 - 0.00342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000868637 + 0.507376i\)
\(L(\frac12)\) \(\approx\) \(0.000868637 + 0.507376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.866 + 1.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.73iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.46 + 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-1.73 + 3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.36329726348468949629465466264, −9.659922424414013512641262954001, −9.110667868648478392076224991342, −8.007801281839146670319788060485, −7.02574838814946377938812159393, −5.46227075140243451906717640114, −4.39531918756213274878655828032, −3.15791342064090526682075058326, −2.42200619049704896762995469668, −0.31149504492441376888974877974, 2.14061573664924536435872093846, 3.20349149156272209553026537353, 5.20055526724219058924563822022, 6.09405800060763669221456629470, 6.98688163923674590744236983753, 7.71225114645679270531125744911, 8.392352664194984708622114505966, 9.257378216202418690141104943126, 9.917365311366034592954295627038, 11.48787581486791090273726941496

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