↑

Properties

Label 2-2100-21.17-c1-0-24
Degree $2$
Conductor $2100$
Sign $0.741 - 0.670i$
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  + (1.69 + 0.368i)3-s + (1.80 − 1.93i)7-s + (2.72 + 1.24i)9-s + (4.05 + 2.34i)11-s + 2.18i·13-s + (−3.74 + 6.49i)17-s + (−0.638 + 0.368i)19-s + (3.76 − 2.61i)21-s + (−6.99 + 4.03i)23-s + (4.15 + 3.11i)27-s − 1.15i·29-s + (8.95 + 5.16i)31-s + (6.00 + 5.45i)33-s + (−2.30 − 3.99i)37-s + (−0.806 + 3.70i)39-s + ⋯
L(s)  = 1  + (0.977 + 0.212i)3-s + (0.680 − 0.732i)7-s + (0.909 + 0.415i)9-s + (1.22 + 0.706i)11-s + 0.607i·13-s + (−0.909 + 1.57i)17-s + (−0.146 + 0.0845i)19-s + (0.820 − 0.571i)21-s + (−1.45 + 0.842i)23-s + (0.800 + 0.599i)27-s − 0.214i·29-s + (1.60 + 0.928i)31-s + (1.04 + 0.950i)33-s + (−0.379 − 0.657i)37-s + (−0.129 + 0.593i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.741 - 0.670i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.741 - 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.891335323\)
\(L(\frac12)\) \(\approx\) \(2.891335323\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.368i)T \)
5 \( 1 \)
7 \( 1 + (-1.80 + 1.93i)T \)
good11 \( 1 + (-4.05 - 2.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.18iT - 13T^{2} \)
17 \( 1 + (3.74 - 6.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.638 - 0.368i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.99 - 4.03i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.15iT - 29T^{2} \)
31 \( 1 + (-8.95 - 5.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.30 + 3.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.43T + 41T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + (4.34 + 7.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.03 + 4.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.48 + 6.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.13 + 2.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.691 - 1.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (-0.211 - 0.122i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.79 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + (-0.658 - 1.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.84iT - 97T^{2} \)
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

−9.101383091736948998706086333935, −8.417545787690125354821025360943, −7.81551932952986070625600303499, −6.90095477178913691890100421514, −6.31353848949291925069335165774, −4.84370923352964708261900838953, −4.05676356237461454309272456847, −3.75001280643326740981960945193, −2.07106842349228417585034679236, −1.54802180175432593176238664564, 0.974079906933383124825489692419, 2.26876869534461527453598169755, 2.91908631508725951248014180530, 4.11163992409668255799579683117, 4.78314424468324339267229514444, 6.05977729966754551119553226307, 6.62583103209321244964455559973, 7.72998534114604305713177080834, 8.260875742302279000113087931787, 8.992275056303339614858817318706

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