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Properties

Label 2-2100-21.11-c0-0-2
Degree $2$
Conductor $2100$
Sign $-0.895 + 0.444i$
Analytic cond. $1.04803$
Root an. cond. $1.02373$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s − 7-s + (−0.499 − 0.866i)9-s − 2·13-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)21-s − 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−1 + 1.73i)39-s + 43-s + 49-s − 1.99·57-s + (0.5 + 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s − 7-s + (−0.499 − 0.866i)9-s − 2·13-s + (−1 − 1.73i)19-s + (−0.5 + 0.866i)21-s − 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−1 + 1.73i)39-s + 43-s + 49-s − 1.99·57-s + (0.5 + 0.866i)61-s + (0.499 + 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(1.04803\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :0),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6982105771\)
\(L(\frac12)\) \(\approx\) \(0.6982105771\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 2T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.119067987565517800567991667680, −8.104620255449959002183226012153, −7.21903574354301994920824613477, −6.87661785654432071081895943124, −6.01035895553357670349764099437, −4.95326137885336850924624301160, −3.92450277127397681537746845221, −2.66605004214087088489473630842, −2.35118307705823380121013576884, −0.41741284568410749007367516442, 2.13866151895462344306578545279, 2.98978676491834756275044514366, 3.86823952812899062202965003619, 4.70786124998326249892450902016, 5.54607648150455333156157480689, 6.48948922826676185148570891508, 7.39306049119319320721627673730, 8.176069391337383464268993776714, 8.957068056411632745930208733391, 9.827936326910205045853000461930

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