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Properties

Label 2-101430-1.1-c1-0-27
Degree $2$
Conductor $101430$
Sign $1$
Analytic cond. $809.922$
Root an. cond. $28.4591$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 4·13-s + 16-s − 2·19-s − 20-s − 23-s + 25-s + 4·26-s − 6·29-s − 2·31-s + 32-s − 10·37-s − 2·38-s − 40-s + 6·41-s − 4·43-s − 46-s + 6·47-s + 50-s + 4·52-s + 6·53-s − 6·58-s + 12·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s + 0.784·26-s − 1.11·29-s − 0.359·31-s + 0.176·32-s − 1.64·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s − 0.147·46-s + 0.875·47-s + 0.141·50-s + 0.554·52-s + 0.824·53-s − 0.787·58-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(101430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(809.922\)
Root analytic conductor: \(28.4591\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 101430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.296792691\)
\(L(\frac12)\) \(\approx\) \(3.296792691\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.52702079683826, −13.34931227568768, −12.88289536513515, −12.20178879521969, −11.91174355365557, −11.35037030165764, −10.81121526262950, −10.57743563212218, −9.890235080509300, −9.203879357745428, −8.687840944372259, −8.273700150027752, −7.679893387365868, −6.975860264284875, −6.817744802476719, −5.917470420418240, −5.638779919953181, −5.056587793438277, −4.286600487013349, −3.808414655025456, −3.536646242461222, −2.695822147437593, −2.038909523131670, −1.386627268295192, −0.5033869740231579, 0.5033869740231579, 1.386627268295192, 2.038909523131670, 2.695822147437593, 3.536646242461222, 3.808414655025456, 4.286600487013349, 5.056587793438277, 5.638779919953181, 5.917470420418240, 6.817744802476719, 6.975860264284875, 7.679893387365868, 8.273700150027752, 8.687840944372259, 9.203879357745428, 9.890235080509300, 10.57743563212218, 10.81121526262950, 11.35037030165764, 11.91174355365557, 12.20178879521969, 12.88289536513515, 13.34931227568768, 13.52702079683826

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