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Properties

Label 2-2100-21.17-c1-0-13
Degree $2$
Conductor $2100$
Sign $0.869 - 0.494i$
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  + (−1.71 + 0.268i)3-s + (−2.57 + 0.615i)7-s + (2.85 − 0.919i)9-s + (−1.80 − 1.04i)11-s − 0.245i·13-s + (0.471 − 0.816i)17-s + (−0.465 + 0.268i)19-s + (4.23 − 1.74i)21-s + (−2.40 + 1.38i)23-s + (−4.63 + 2.34i)27-s − 0.267i·29-s + (0.981 + 0.566i)31-s + (3.37 + 1.29i)33-s + (−3.08 − 5.33i)37-s + (0.0660 + 0.420i)39-s + ⋯
L(s)  = 1  + (−0.987 + 0.155i)3-s + (−0.972 + 0.232i)7-s + (0.951 − 0.306i)9-s + (−0.544 − 0.314i)11-s − 0.0681i·13-s + (0.114 − 0.198i)17-s + (−0.106 + 0.0616i)19-s + (0.924 − 0.380i)21-s + (−0.500 + 0.288i)23-s + (−0.892 + 0.450i)27-s − 0.0496i·29-s + (0.176 + 0.101i)31-s + (0.586 + 0.226i)33-s + (−0.506 − 0.877i)37-s + (0.0105 + 0.0673i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.869 - 0.494i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8188400333\)
\(L(\frac12)\) \(\approx\) \(0.8188400333\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.268i)T \)
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

−9.317649364940164026792360659920, −8.468534522106207942361429893595, −7.37017921357408377521589548857, −6.79280706137224122929104104302, −5.78444075913946110749279684301, −5.50363689693437308235868179333, −4.30462290754045613540772864185, −3.47354428385994503988981889810, −2.30277434271174751482816592049, −0.69619299018534766959186737287, 0.53570966725699453000463947262, 1.98847958479489377565244833122, 3.24063843320667169841869655567, 4.28320583598925435609043753295, 5.07718319869850560704637881877, 6.03919786007023693583009536511, 6.53645276924896910361191731874, 7.36666301242472182846568605268, 8.086494870900058554021021373878, 9.249949778424834111381715325251

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