↑

Properties

Label 2-3525-141.50-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.949 - 0.314i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
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  + (−0.211 − 0.347i)2-s + (−0.816 − 0.576i)3-s + (0.383 − 0.740i)4-s + (−0.0277 + 0.405i)6-s + (−0.744 + 0.0509i)8-s + (0.334 + 0.942i)9-s + (−0.740 + 0.383i)12-s + (−0.306 − 0.433i)16-s + (0.262 + 1.90i)17-s + (0.256 − 0.315i)18-s + (−0.187 + 0.900i)19-s + (−0.887 + 1.46i)23-s + (0.637 + 0.387i)24-s + (0.269 − 0.962i)27-s + (1.14 + 1.61i)31-s + (−0.383 + 0.882i)32-s + ⋯
L(s)  = 1  + (−0.211 − 0.347i)2-s + (−0.816 − 0.576i)3-s + (0.383 − 0.740i)4-s + (−0.0277 + 0.405i)6-s + (−0.744 + 0.0509i)8-s + (0.334 + 0.942i)9-s + (−0.740 + 0.383i)12-s + (−0.306 − 0.433i)16-s + (0.262 + 1.90i)17-s + (0.256 − 0.315i)18-s + (−0.187 + 0.900i)19-s + (−0.887 + 1.46i)23-s + (0.637 + 0.387i)24-s + (0.269 − 0.962i)27-s + (1.14 + 1.61i)31-s + (−0.383 + 0.882i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6690469904\)
\(L(\frac12)\) \(\approx\) \(0.6690469904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.816 + 0.576i)T \)
5 \( 1 \)
47 \( 1 + (0.730 + 0.682i)T \)
good2 \( 1 + (0.211 + 0.347i)T + (-0.460 + 0.887i)T^{2} \)
7 \( 1 + (0.854 - 0.519i)T^{2} \)
11 \( 1 + (-0.962 - 0.269i)T^{2} \)
13 \( 1 + (0.203 - 0.979i)T^{2} \)
17 \( 1 + (-0.262 - 1.90i)T + (-0.962 + 0.269i)T^{2} \)
19 \( 1 + (0.187 - 0.900i)T + (-0.917 - 0.398i)T^{2} \)
23 \( 1 + (0.887 - 1.46i)T + (-0.460 - 0.887i)T^{2} \)
29 \( 1 + (-0.203 - 0.979i)T^{2} \)
31 \( 1 + (-1.14 - 1.61i)T + (-0.334 + 0.942i)T^{2} \)
37 \( 1 + (-0.0682 + 0.997i)T^{2} \)
41 \( 1 + (0.990 + 0.136i)T^{2} \)
43 \( 1 + (-0.576 - 0.816i)T^{2} \)
53 \( 1 + (-0.668 - 0.0457i)T + (0.990 + 0.136i)T^{2} \)
59 \( 1 + (0.576 - 0.816i)T^{2} \)
61 \( 1 + (1.05 - 1.13i)T + (-0.0682 - 0.997i)T^{2} \)
67 \( 1 + (0.854 + 0.519i)T^{2} \)
71 \( 1 + (-0.460 - 0.887i)T^{2} \)
73 \( 1 + (-0.775 - 0.631i)T^{2} \)
79 \( 1 + (1.81 + 0.789i)T + (0.682 + 0.730i)T^{2} \)
83 \( 1 + (0.0185 - 0.135i)T + (-0.962 - 0.269i)T^{2} \)
89 \( 1 + (0.917 - 0.398i)T^{2} \)
97 \( 1 + (-0.334 - 0.942i)T^{2} \)
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

−8.692339702299148628734638586458, −8.047148258131591999367225725540, −7.21514455884523247949319175334, −6.33270908562748921934648857949, −5.94090444286428534100505001801, −5.28543048926965712260932039395, −4.22145276524274306412899013735, −3.12905366702151958065919371963, −1.76562713877241689338123496642, −1.42071375293452231198592124215, 0.46789341032515179143221586533, 2.41962691408262783149047650945, 3.17314716025474290245285825660, 4.30240375549558827601063305729, 4.80883009856506272105221329192, 5.87990092234065702499315815623, 6.52688328771823380160720048787, 7.11356415281730534000008728564, 7.928838262958515400136112152360, 8.688289857934221632399098594863

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