Properties

Label 2-2100-105.59-c1-0-0
Degree $2$
Conductor $2100$
Sign $-0.916 + 0.399i$
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  + (−1.56 + 0.747i)3-s + (−2.52 + 0.786i)7-s + (1.88 − 2.33i)9-s + (−2.34 − 1.35i)11-s + 1.12·13-s + (6.39 + 3.69i)17-s + (0.412 − 0.238i)19-s + (3.35 − 3.11i)21-s + (2.79 + 4.84i)23-s + (−1.19 + 5.05i)27-s − 2.20i·29-s + (−2.07 − 1.19i)31-s + (4.67 + 0.362i)33-s + (−7.53 + 4.34i)37-s + (−1.75 + 0.840i)39-s + ⋯
L(s)  = 1  + (−0.902 + 0.431i)3-s + (−0.954 + 0.297i)7-s + (0.627 − 0.778i)9-s + (−0.706 − 0.408i)11-s + 0.311·13-s + (1.55 + 0.895i)17-s + (0.0945 − 0.0546i)19-s + (0.732 − 0.680i)21-s + (0.583 + 1.01i)23-s + (−0.230 + 0.973i)27-s − 0.409i·29-s + (−0.372 − 0.214i)31-s + (0.813 + 0.0631i)33-s + (−1.23 + 0.714i)37-s + (−0.281 + 0.134i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.916 + 0.399i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06338121371\)
\(L(\frac12)\) \(\approx\) \(0.06338121371\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 - 0.747i)T \)
5 \( 1 \)
7 \( 1 + (2.52 - 0.786i)T \)
good11 \( 1 + (2.34 + 1.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.12T + 13T^{2} \)
17 \( 1 + (-6.39 - 3.69i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.412 + 0.238i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.79 - 4.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.20iT - 29T^{2} \)
31 \( 1 + (2.07 + 1.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.53 - 4.34i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.42T + 41T^{2} \)
43 \( 1 + 4.16iT - 43T^{2} \)
47 \( 1 + (10.7 - 6.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.45 + 4.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.15 + 2.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.26 - 1.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.7 + 7.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.89iT - 71T^{2} \)
73 \( 1 + (-3.63 + 6.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.47 + 6.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (-3.48 - 6.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.79T + 97T^{2} \)
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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

−9.757615315956344346793566232599, −8.899985831665820678911689587929, −7.974628397815434202162560437637, −7.07291815858862463963097330861, −6.18782977813411955910847732743, −5.63689840634247797705067057725, −4.96177870707215891735763261364, −3.61813612592981940745396452198, −3.20282532354954201136484977098, −1.45087198587592600692705843078, 0.02793448542678938424895562257, 1.26681456404031961522921983654, 2.69305183654520101827864679217, 3.64019428933676745965771191345, 4.90429952960177657096158282471, 5.43587524788405344360149915010, 6.37314750544115937908173888813, 7.08209116300713098227885453049, 7.60395967504398373109796565327, 8.624361457450487947418783696321

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