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Properties

Label 2-2100-105.59-c1-0-30
Degree $2$
Conductor $2100$
Sign $0.955 + 0.293i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 + 0.404i)3-s + (−2.10 − 1.60i)7-s + (2.67 + 1.36i)9-s + (−2.05 − 1.18i)11-s + 0.748·13-s + (6.53 + 3.77i)17-s + (6.11 − 3.53i)19-s + (−2.88 − 3.55i)21-s + (−1.63 − 2.83i)23-s + (3.95 + 3.37i)27-s + 2.48i·29-s + (−6.84 − 3.95i)31-s + (−2.98 − 2.83i)33-s + (−3.73 + 2.15i)37-s + (1.26 + 0.302i)39-s + ⋯
L(s)  = 1  + (0.972 + 0.233i)3-s + (−0.794 − 0.607i)7-s + (0.891 + 0.453i)9-s + (−0.620 − 0.358i)11-s + 0.207·13-s + (1.58 + 0.914i)17-s + (1.40 − 0.810i)19-s + (−0.630 − 0.776i)21-s + (−0.340 − 0.590i)23-s + (0.760 + 0.649i)27-s + 0.461i·29-s + (−1.22 − 0.709i)31-s + (−0.519 − 0.493i)33-s + (−0.614 + 0.354i)37-s + (0.201 + 0.0484i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.436843330\)
\(L(\frac12)\) \(\approx\) \(2.436843330\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.404i)T \)
5 \( 1 \)
7 \( 1 + (2.10 + 1.60i)T \)
good11 \( 1 + (2.05 + 1.18i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.748T + 13T^{2} \)
17 \( 1 + (-6.53 - 3.77i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.11 + 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.63 + 2.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.48iT - 29T^{2} \)
31 \( 1 + (6.84 + 3.95i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.73 - 2.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 3.03iT - 43T^{2} \)
47 \( 1 + (-5.59 + 3.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0540 + 0.0935i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.60 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 + 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.09 - 2.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-0.780 + 1.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.27 - 2.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.901iT - 83T^{2} \)
89 \( 1 + (-2.43 - 4.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.190531072652326928594585871666, −8.210317804281134241022657049288, −7.62507959795392829765274826565, −6.96652373652997447960227046963, −5.85412582634433697807235865266, −5.03539572539453645030957236844, −3.73950602498961974466863159441, −3.42478620250741970739163295015, −2.35664948476147957165732428740, −0.915661526217939627035710160558, 1.16658159031247204223694750075, 2.47123579275808581432478133905, 3.19603575960525338226532189573, 3.92469260836273127511200059077, 5.37232914128048974273762815012, 5.81705403739886034993053936194, 7.22671392315438251240599099095, 7.45680226754130088577051138806, 8.349092223204394681497374508250, 9.260312001639549376613949048941

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