↑

Properties

Label 2-525-35.3-c1-0-13
Degree $2$
Conductor $525$
Sign $-0.153 + 0.988i$
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  + (−0.595 − 0.159i)2-s + (−0.258 − 0.965i)3-s + (−1.40 − 0.810i)4-s + 0.616i·6-s + (2.22 + 1.43i)7-s + (1.57 + 1.57i)8-s + (−0.866 + 0.499i)9-s + (2.27 − 3.94i)11-s + (−0.419 + 1.56i)12-s + (0.0478 − 0.0478i)13-s + (−1.09 − 1.20i)14-s + (0.932 + 1.61i)16-s + (4.03 − 1.07i)17-s + (0.595 − 0.159i)18-s + (−3.38 − 5.86i)19-s + ⋯
L(s)  = 1  + (−0.420 − 0.112i)2-s + (−0.149 − 0.557i)3-s + (−0.701 − 0.405i)4-s + 0.251i·6-s + (0.840 + 0.542i)7-s + (0.557 + 0.557i)8-s + (−0.288 + 0.166i)9-s + (0.686 − 1.18i)11-s + (−0.121 + 0.451i)12-s + (0.0132 − 0.0132i)13-s + (−0.292 − 0.323i)14-s + (0.233 + 0.403i)16-s + (0.977 − 0.261i)17-s + (0.140 − 0.0376i)18-s + (−0.776 − 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.153 + 0.988i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617635 - 0.721225i\)
\(L(\frac12)\) \(\approx\) \(0.617635 - 0.721225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.22 - 1.43i)T \)
good2 \( 1 + (0.595 + 0.159i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-2.27 + 3.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0478 + 0.0478i)T - 13iT^{2} \)
17 \( 1 + (-4.03 + 1.07i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.38 + 5.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.05 - 3.94i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.68iT - 29T^{2} \)
31 \( 1 + (2.56 + 1.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.13 + 1.91i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.956iT - 41T^{2} \)
43 \( 1 + (3.47 + 3.47i)T + 43iT^{2} \)
47 \( 1 + (-2.35 + 8.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-12.1 + 3.24i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.09 + 1.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.569 + 0.329i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.72 - 10.1i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.58T + 71T^{2} \)
73 \( 1 + (-1.98 - 7.42i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.84 + 2.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.19 + 7.19i)T - 83iT^{2} \)
89 \( 1 + (-7.86 - 13.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.60 + 4.60i)T + 97iT^{2} \)
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

−10.71524549932066218047694436116, −9.591735157162067497610991321401, −8.691054576125505956571160404614, −8.260961236397634475983348873299, −7.08547228089488267657588496489, −5.79113337975241345101226020652, −5.21430736770490022275545172029, −3.84240634063219197188778129264, −2.11529441988666680819181546623, −0.75035408999259853570125943233, 1.50655931889399607000422968719, 3.65448219979887381066775153245, 4.36815709014707372388470053639, 5.25347469477739661095185796342, 6.73075005366023598419088950025, 7.72661254259237287880301014164, 8.443337579705902853728533618928, 9.356178251091920833862229304260, 10.22660502171495953911860056638, 10.71333639980158303834463573898

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