Properties

Label 2-2205-315.304-c0-0-0
Degree $2$
Conductor $2205$
Sign $-0.281 - 0.959i$
Analytic cond. $1.10043$
Root an. cond. $1.04901$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)20-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (−1 + 1.73i)29-s − 0.999·33-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)20-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (−1 + 1.73i)29-s − 0.999·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.281 - 0.959i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.281 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167261605\)
\(L(\frac12)\) \(\approx\) \(1.167261605\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 - T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.617165939713496771234483052794, −8.873277321730015430779078568338, −7.67328203531851811377422707575, −6.96699070314400794099360902047, −6.51989606614620395412128172753, −5.58948473666710481326945648670, −4.59234477865635825934491422907, −3.65357684221529865691539372548, −3.00316903791872139917304019375, −1.69606744992959541903140378407, 0.900154250872804777173279294693, 1.82916672696701128280875576354, 3.11461549321011374022210516117, 4.03672615170658931443114839217, 5.44115461187505276592197455638, 5.87904768139759362611052414295, 6.57620383145966869357765757476, 7.63618700609401585382443780578, 8.034568816534732898264124423411, 8.668346314552491140691315574926

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