Properties

Label 2-2100-5.4-c1-0-15
Degree $2$
Conductor $2100$
Sign $-0.447 + 0.894i$
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  + i·3-s + i·7-s − 9-s − 2·11-s − 4i·13-s + 2i·17-s − 2·19-s − 21-s − 4i·23-s i·27-s − 6·29-s − 2·31-s − 2i·33-s + 10i·37-s + 4·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 0.603·11-s − 1.10i·13-s + 0.485i·17-s − 0.458·19-s − 0.218·21-s − 0.834i·23-s − 0.192i·27-s − 1.11·29-s − 0.359·31-s − 0.348i·33-s + 1.64i·37-s + 0.640·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4378640517\)
\(L(\frac12)\) \(\approx\) \(0.4378640517\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.621618392614239873815125769025, −8.383656814787692730998199955707, −7.36220778775770456810332207816, −6.39179153417117371018808781593, −5.49725271017642337833106955652, −4.97708332450268880609183935520, −3.84165440582506064484745163952, −3.03461871075041817737562240181, −1.97217767069043706481230098516, −0.14720698211932610698994654897, 1.44825648772511851006022703499, 2.41788727084920350811030816656, 3.56450321801565425881537584698, 4.50309858388751131446906466727, 5.46204864132805778753100852409, 6.27914087527691853401387882290, 7.20627459405208267087787105295, 7.57368372945139556058656651491, 8.577242328406466319352978760858, 9.301937215550944071328515025940

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