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Properties

Label 2-2100-21.17-c1-0-16
Degree $2$
Conductor $2100$
Sign $0.666 + 0.745i$
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  + (−0.622 − 1.61i)3-s + (−2.57 + 0.615i)7-s + (−2.22 + 2.01i)9-s + (1.80 + 1.04i)11-s − 0.245i·13-s + (−0.471 + 0.816i)17-s + (−0.465 + 0.268i)19-s + (2.59 + 3.77i)21-s + (2.40 − 1.38i)23-s + (4.63 + 2.34i)27-s + 0.267i·29-s + (0.981 + 0.566i)31-s + (0.560 − 3.56i)33-s + (−3.08 − 5.33i)37-s + (−0.397 + 0.153i)39-s + ⋯
L(s)  = 1  + (−0.359 − 0.933i)3-s + (−0.972 + 0.232i)7-s + (−0.741 + 0.671i)9-s + (0.544 + 0.314i)11-s − 0.0681i·13-s + (−0.114 + 0.198i)17-s + (−0.106 + 0.0616i)19-s + (0.566 + 0.823i)21-s + (0.500 − 0.288i)23-s + (0.892 + 0.450i)27-s + 0.0496i·29-s + (0.176 + 0.101i)31-s + (0.0975 − 0.621i)33-s + (−0.506 − 0.877i)37-s + (−0.0636 + 0.0245i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.666 + 0.745i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.226318577\)
\(L(\frac12)\) \(\approx\) \(1.226318577\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.622 + 1.61i)T \)
5 \( 1 \)
7 \( 1 + (2.57 - 0.615i)T \)
good11 \( 1 + (-1.80 - 1.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.245iT - 13T^{2} \)
17 \( 1 + (0.471 - 0.816i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.465 - 0.268i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.40 + 1.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.267iT - 29T^{2} \)
31 \( 1 + (-0.981 - 0.566i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.08 + 5.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.38T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (-6.23 - 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.8 + 6.26i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (2.78 - 4.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (-11.3 - 6.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.17 - 5.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.06T + 83T^{2} \)
89 \( 1 + (-0.463 - 0.803i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.01iT - 97T^{2} \)
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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

−9.039566690476653012104721395578, −8.132064156861433246703394094808, −7.30329703200314911148405487403, −6.61314863565555780445306625820, −6.05667926589206223475129531600, −5.19299525634382545998120576990, −4.04569700800878264655166294307, −2.95534566755713612874058343220, −2.01708572460236632740536121028, −0.68737450260328778298243226623, 0.78629608517346190130845661429, 2.64596227467788769030280640255, 3.55922077619732168337925162437, 4.21766574359881447799649685896, 5.21418797639838527335192113875, 6.05694764057139141121328900029, 6.67097189340642253177529198862, 7.59184725191989520237039090840, 8.861825736004289422813196851530, 9.140405169494148398391461501631

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