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Properties

Label 2-2100-35.33-c1-0-21
Degree $2$
Conductor $2100$
Sign $-0.753 + 0.657i$
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  + (0.965 + 0.258i)3-s + (−2.25 + 1.38i)7-s + (0.866 + 0.499i)9-s + (−2.42 − 4.19i)11-s + (−2.27 + 2.27i)13-s + (−0.574 + 2.14i)17-s + (−0.107 + 0.185i)19-s + (−2.53 + 0.758i)21-s + (6.64 − 1.78i)23-s + (0.707 + 0.707i)27-s − 9.28i·29-s + (−7.78 + 4.49i)31-s + (−1.25 − 4.68i)33-s + (−2.62 − 9.78i)37-s + (−2.78 + 1.61i)39-s + ⋯
L(s)  = 1  + (0.557 + 0.149i)3-s + (−0.851 + 0.524i)7-s + (0.288 + 0.166i)9-s + (−0.730 − 1.26i)11-s + (−0.631 + 0.631i)13-s + (−0.139 + 0.520i)17-s + (−0.0246 + 0.0426i)19-s + (−0.553 + 0.165i)21-s + (1.38 − 0.371i)23-s + (0.136 + 0.136i)27-s − 1.72i·29-s + (−1.39 + 0.807i)31-s + (−0.218 − 0.815i)33-s + (−0.431 − 1.60i)37-s + (−0.446 + 0.257i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4518950337\)
\(L(\frac12)\) \(\approx\) \(0.4518950337\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.25 - 1.38i)T \)
good11 \( 1 + (2.42 + 4.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.27 - 2.27i)T - 13iT^{2} \)
17 \( 1 + (0.574 - 2.14i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.107 - 0.185i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.64 + 1.78i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.28iT - 29T^{2} \)
31 \( 1 + (7.78 - 4.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.62 + 9.78i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.32iT - 41T^{2} \)
43 \( 1 + (5.01 + 5.01i)T + 43iT^{2} \)
47 \( 1 + (9.36 - 2.50i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.785 - 2.93i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.68 + 4.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.2 + 7.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.97 - 1.33i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (-0.146 - 0.0391i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.0599 + 0.0346i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.467 + 0.467i)T - 83iT^{2} \)
89 \( 1 + (0.997 - 1.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.51 - 7.51i)T + 97iT^{2} \)
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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

−8.943882854988360230983269485426, −8.152731230613810407499656342388, −7.30098049253141812843101386435, −6.43173245512855634831854125386, −5.66230865605605050103873846462, −4.77321962745982578529788275324, −3.60322152041875601791900286630, −2.96173197082517929432327530729, −1.99780586219474428086196733018, −0.13727979834743880588847154716, 1.54937179421926105736364480762, 2.82455682208335374705452783633, 3.35600422426523084780572560409, 4.66351086088492802555365175894, 5.21666505411971761978600329772, 6.52452008146352605874362859573, 7.32384694305189997292116461609, 7.52203141916160648659562575908, 8.736657195929885470172389660657, 9.433312203134912676154635389947

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