Properties

Label 2-2100-5.4-c3-0-8
Degree $2$
Conductor $2100$
Sign $0.447 - 0.894i$
Analytic cond. $123.904$
Root an. cond. $11.1312$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 7i·7-s − 9·9-s + 4·11-s − 54i·13-s − 14i·17-s − 92·19-s − 21·21-s + 152i·23-s + 27i·27-s + 106·29-s − 144·31-s − 12i·33-s + 158i·37-s − 162·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.109·11-s − 1.15i·13-s − 0.199i·17-s − 1.11·19-s − 0.218·21-s + 1.37i·23-s + 0.192i·27-s + 0.678·29-s − 0.834·31-s − 0.0633i·33-s + 0.702i·37-s − 0.665·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(123.904\)
Root analytic conductor: \(11.1312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8980327255\)
\(L(\frac12)\) \(\approx\) \(0.8980327255\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good11 \( 1 - 4T + 1.33e3T^{2} \)
13 \( 1 + 54iT - 2.19e3T^{2} \)
17 \( 1 + 14iT - 4.91e3T^{2} \)
19 \( 1 + 92T + 6.85e3T^{2} \)
23 \( 1 - 152iT - 1.21e4T^{2} \)
29 \( 1 - 106T + 2.43e4T^{2} \)
31 \( 1 + 144T + 2.97e4T^{2} \)
37 \( 1 - 158iT - 5.06e4T^{2} \)
41 \( 1 + 390T + 6.89e4T^{2} \)
43 \( 1 - 508iT - 7.95e4T^{2} \)
47 \( 1 + 528iT - 1.03e5T^{2} \)
53 \( 1 + 606iT - 1.48e5T^{2} \)
59 \( 1 - 364T + 2.05e5T^{2} \)
61 \( 1 - 678T + 2.26e5T^{2} \)
67 \( 1 - 844iT - 3.00e5T^{2} \)
71 \( 1 + 8T + 3.57e5T^{2} \)
73 \( 1 - 422iT - 3.89e5T^{2} \)
79 \( 1 + 384T + 4.93e5T^{2} \)
83 \( 1 - 548iT - 5.71e5T^{2} \)
89 \( 1 + 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + 1.50e3iT - 9.12e5T^{2} \)
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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

−8.542222737489697285052910572286, −8.245525450570863198102638050201, −7.23360785145816925042800614774, −6.70870919036987273729668193862, −5.71584198271652716078803442801, −5.03452139139164025086740393084, −3.85464774016382354694199052601, −3.03862808860091806089384428942, −1.91986023388424447428465672186, −0.893914474519227313754950317785, 0.20740427915651909944462589503, 1.78937730751274217184649329827, 2.64518240234891050233234077620, 3.86524136341126907758285643072, 4.45315358010991380576863299478, 5.34907356905472767499163831716, 6.32606465557654087179360601218, 6.86202624460057260219195230945, 8.014987727968908900509961073315, 8.873710292611585553740468772235

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