Properties

Label 2-2100-105.89-c1-0-13
Degree $2$
Conductor $2100$
Sign $0.830 - 0.556i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
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  + (0.268 − 1.71i)3-s + (−0.615 + 2.57i)7-s + (−2.85 − 0.919i)9-s + (−1.80 + 1.04i)11-s − 0.245·13-s + (0.816 − 0.471i)17-s + (0.465 + 0.268i)19-s + (4.23 + 1.74i)21-s + (1.38 − 2.40i)23-s + (−2.34 + 4.63i)27-s − 0.267i·29-s + (0.981 − 0.566i)31-s + (1.29 + 3.37i)33-s + (5.33 + 3.08i)37-s + (−0.0660 + 0.420i)39-s + ⋯
L(s)  = 1  + (0.155 − 0.987i)3-s + (−0.232 + 0.972i)7-s + (−0.951 − 0.306i)9-s + (−0.544 + 0.314i)11-s − 0.0681·13-s + (0.198 − 0.114i)17-s + (0.106 + 0.0616i)19-s + (0.924 + 0.380i)21-s + (0.288 − 0.500i)23-s + (−0.450 + 0.892i)27-s − 0.0496i·29-s + (0.176 − 0.101i)31-s + (0.226 + 0.586i)33-s + (0.877 + 0.506i)37-s + (−0.0105 + 0.0673i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.830 - 0.556i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.363780454\)
\(L(\frac12)\) \(\approx\) \(1.363780454\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.268 + 1.71i)T \)
5 \( 1 \)
7 \( 1 + (0.615 - 2.57i)T \)
good11 \( 1 + (1.80 - 1.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.245T + 13T^{2} \)
17 \( 1 + (-0.816 + 0.471i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.465 - 0.268i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.38 + 2.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.267iT - 29T^{2} \)
31 \( 1 + (-0.981 + 0.566i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.33 - 3.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.38T + 41T^{2} \)
43 \( 1 - 11.4iT - 43T^{2} \)
47 \( 1 + (-10.7 - 6.23i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.26 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.25 - 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.96 - 2.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.81 - 2.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (-6.56 - 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.17 - 5.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.06iT - 83T^{2} \)
89 \( 1 + (-0.463 + 0.803i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.01T + 97T^{2} \)
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

−9.051500920008923645150435879687, −8.330126758706469881869458329774, −7.65751295843919884632381264339, −6.87655332949934636261829020062, −6.02176370494716866861299939044, −5.45781465786547570500545305697, −4.34061760611212062625793017806, −2.88199065724407845627626235992, −2.48074491551869442569644644196, −1.13812634738545084340443505232, 0.52288337189313621413686337925, 2.30232863467395176332745866233, 3.45272822519654759004452997183, 3.95522585663364939426068604542, 5.02109739252983934577427098741, 5.62643094062154014775120928401, 6.73732153125621732577298856396, 7.55765809517606281691731864607, 8.345126460845817944796691909402, 9.098024400123183762375821715312

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