↑

Properties

Label 2-3525-705.107-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.921 - 0.388i$
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  + (−1.34 − 0.231i)2-s + (−0.994 − 0.102i)3-s + (0.813 + 0.289i)4-s + (1.31 + 0.368i)6-s + (0.162 + 0.0913i)8-s + (0.979 + 0.203i)9-s + (−0.779 − 0.370i)12-s + (−0.867 − 0.705i)16-s + (0.217 − 0.893i)17-s + (−1.26 − 0.500i)18-s + (1.28 + 1.37i)19-s + (0.277 + 1.61i)23-s + (−0.152 − 0.107i)24-s + (−0.953 − 0.302i)27-s + (0.655 − 0.806i)31-s + (0.880 + 1.00i)32-s + ⋯
L(s)  = 1  + (−1.34 − 0.231i)2-s + (−0.994 − 0.102i)3-s + (0.813 + 0.289i)4-s + (1.31 + 0.368i)6-s + (0.162 + 0.0913i)8-s + (0.979 + 0.203i)9-s + (−0.779 − 0.370i)12-s + (−0.867 − 0.705i)16-s + (0.217 − 0.893i)17-s + (−1.26 − 0.500i)18-s + (1.28 + 1.37i)19-s + (0.277 + 1.61i)23-s + (−0.152 − 0.107i)24-s + (−0.953 − 0.302i)27-s + (0.655 − 0.806i)31-s + (0.880 + 1.00i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.921 - 0.388i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.921 - 0.388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4283904602\)
\(L(\frac12)\) \(\approx\) \(0.4283904602\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.994 + 0.102i)T \)
5 \( 1 \)
47 \( 1 + (-0.604 + 0.796i)T \)
good2 \( 1 + (1.34 + 0.231i)T + (0.942 + 0.334i)T^{2} \)
7 \( 1 + (0.816 - 0.576i)T^{2} \)
11 \( 1 + (0.460 + 0.887i)T^{2} \)
13 \( 1 + (-0.730 + 0.682i)T^{2} \)
17 \( 1 + (-0.217 + 0.893i)T + (-0.887 - 0.460i)T^{2} \)
19 \( 1 + (-1.28 - 1.37i)T + (-0.0682 + 0.997i)T^{2} \)
23 \( 1 + (-0.277 - 1.61i)T + (-0.942 + 0.334i)T^{2} \)
29 \( 1 + (-0.682 + 0.730i)T^{2} \)
31 \( 1 + (-0.655 + 0.806i)T + (-0.203 - 0.979i)T^{2} \)
37 \( 1 + (-0.269 + 0.962i)T^{2} \)
41 \( 1 + (0.854 + 0.519i)T^{2} \)
43 \( 1 + (0.631 - 0.775i)T^{2} \)
53 \( 1 + (-0.199 - 0.354i)T + (-0.519 + 0.854i)T^{2} \)
59 \( 1 + (-0.775 + 0.631i)T^{2} \)
61 \( 1 + (1.81 - 0.249i)T + (0.962 - 0.269i)T^{2} \)
67 \( 1 + (0.816 + 0.576i)T^{2} \)
71 \( 1 + (0.334 + 0.942i)T^{2} \)
73 \( 1 + (-0.398 - 0.917i)T^{2} \)
79 \( 1 + (1.70 + 0.116i)T + (0.990 + 0.136i)T^{2} \)
83 \( 1 + (-0.455 - 1.87i)T + (-0.887 + 0.460i)T^{2} \)
89 \( 1 + (-0.0682 - 0.997i)T^{2} \)
97 \( 1 + (0.979 + 0.203i)T^{2} \)
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

−9.014785505857152581323305518924, −7.78300218857848429358369081638, −7.66001251772779311189510819889, −6.82207604809567405372485854734, −5.77182347255109784858613423515, −5.24453075017274904416340518727, −4.25514414287487619915757234220, −3.08709005508049848345909583834, −1.74079593417524690861907984167, −0.970609857229628841994214450717, 0.62687615900979504538100264221, 1.61723265466847604698527931147, 3.01885263542065907829081293242, 4.35134905037514542121645686171, 4.90712109851293834755589326560, 5.97334231561146640087496356260, 6.68149745366251930923744108106, 7.22547291929230369440277090824, 8.003222153427286136037312564087, 8.800416784689805862666563845238

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