Properties

Label 2-28566-1.1-c1-0-10
Degree $2$
Conductor $28566$
Sign $-1$
Analytic cond. $228.100$
Root an. cond. $15.1030$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 3·11-s − 4·13-s − 14-s + 16-s − 2·19-s − 3·20-s − 3·22-s + 4·25-s + 4·26-s + 28-s + 6·29-s + 5·31-s − 32-s − 3·35-s − 2·37-s + 2·38-s + 3·40-s − 6·41-s + 10·43-s + 3·44-s + 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.670·20-s − 0.639·22-s + 4/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s + 0.898·31-s − 0.176·32-s − 0.507·35-s − 0.328·37-s + 0.324·38-s + 0.474·40-s − 0.937·41-s + 1.52·43-s + 0.452·44-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28566\)    =    \(2 \cdot 3^{3} \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(228.100\)
Root analytic conductor: \(15.1030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28566,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.51731793770337, −15.05887791208402, −14.45711489610508, −14.18036486154823, −13.27570413810657, −12.50361610202890, −12.01154234475471, −11.81059881045570, −11.26155004513572, −10.52339100895060, −10.18677451939172, −9.370297569176822, −8.873315899420236, −8.346078210322909, −7.733331211376820, −7.426786857726018, −6.691302078715823, −6.255131964865060, −5.258525174349598, −4.490901924937799, −4.188660563979319, −3.266758360267214, −2.658009140212916, −1.728618804938094, −0.8595831211664469, 0, 0.8595831211664469, 1.728618804938094, 2.658009140212916, 3.266758360267214, 4.188660563979319, 4.490901924937799, 5.258525174349598, 6.255131964865060, 6.691302078715823, 7.426786857726018, 7.733331211376820, 8.346078210322909, 8.873315899420236, 9.370297569176822, 10.18677451939172, 10.52339100895060, 11.26155004513572, 11.81059881045570, 12.01154234475471, 12.50361610202890, 13.27570413810657, 14.18036486154823, 14.45711489610508, 15.05887791208402, 15.51731793770337

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