↑

Properties

Label 2-525-7.2-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.922 + 0.386i$
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  + (−1.36 + 2.36i)2-s + (−0.5 − 0.866i)3-s + (−2.73 − 4.73i)4-s + 2.73·6-s + (−0.866 − 2.5i)7-s + 9.46·8-s + (−0.499 + 0.866i)9-s + (−0.366 − 0.633i)11-s + (−2.73 + 4.73i)12-s − 2.26·13-s + (7.09 + 1.36i)14-s + (−7.46 + 12.9i)16-s + (1.63 + 2.83i)17-s + (−1.36 − 2.36i)18-s + (−2.23 + 3.86i)19-s + ⋯
L(s)  = 1  + (−0.965 + 1.67i)2-s + (−0.288 − 0.499i)3-s + (−1.36 − 2.36i)4-s + 1.11·6-s + (−0.327 − 0.944i)7-s + 3.34·8-s + (−0.166 + 0.288i)9-s + (−0.110 − 0.191i)11-s + (−0.788 + 1.36i)12-s − 0.629·13-s + (1.89 + 0.365i)14-s + (−1.86 + 3.23i)16-s + (0.396 + 0.686i)17-s + (−0.321 − 0.557i)18-s + (−0.512 + 0.886i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.922 + 0.386i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0359668 - 0.178877i\)
\(L(\frac12)\) \(\approx\) \(0.0359668 - 0.178877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (1.36 - 2.36i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.366 + 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 + (-1.63 - 2.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.23 - 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 + (-0.232 - 0.401i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.59 - 2.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.732T + 41T^{2} \)
43 \( 1 + 3.19T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.19 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0980 - 0.169i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.33 + 12.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + (-6.33 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.69 + 6.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (7.56 - 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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

−10.87681671641877793249693891816, −10.18801416546287949801815846011, −9.476314684838113692070540040026, −8.303779396959485159947060209854, −7.68703974669635499769986208650, −6.98123974872391439139782675325, −6.12767522862625567105627362265, −5.35336596357687700716879020563, −4.04885955873784853450992779165, −1.43164456206703976689329821328, 0.15905344153980337818924475531, 2.16602590777935336527781732797, 3.02780322225673365980808133131, 4.26160705236712672132013570803, 5.29895431281522139939526090243, 6.97672055338985661064281058813, 8.197925406214220725868553583055, 8.998476057543805380668524343476, 9.647715086344724749135802758872, 10.28594571085522490486337334775

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