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Properties

Label 2-101430-1.1-c1-0-24
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 − 2·11-s − 2·13-s + 16-s + 2·17-s + 4·19-s − 20-s − 2·22-s + 23-s + 25-s − 2·26-s − 8·29-s − 2·31-s + 32-s + 2·34-s + 4·37-s + 4·38-s − 40-s + 2·41-s − 8·43-s − 2·44-s + 46-s + 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s − 1.48·29-s − 0.359·31-s + 0.176·32-s + 0.342·34-s + 0.657·37-s + 0.648·38-s − 0.158·40-s + 0.312·41-s − 1.21·43-s − 0.301·44-s + 0.147·46-s + 1.75·47-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

\(L(1)\) \(\approx\) \(2.793138177\)
\(L(\frac12)\) \(\approx\) \(2.793138177\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.52116660766980, −13.41881340116052, −12.70327014564062, −12.33057500256363, −11.89892141755613, −11.33896493803202, −10.92029381615280, −10.46516902470325, −9.714960095797038, −9.463161614696793, −8.752546163450167, −8.027641022944459, −7.616898883970156, −7.289544571077136, −6.719881205157511, −5.855047841826159, −5.640389513272712, −4.885821218833554, −4.625840318312028, −3.683575868400398, −3.444773589889960, −2.711893670515226, −2.137555311384648, −1.338476312094239, −0.4646031856104769, 0.4646031856104769, 1.338476312094239, 2.137555311384648, 2.711893670515226, 3.444773589889960, 3.683575868400398, 4.625840318312028, 4.885821218833554, 5.640389513272712, 5.855047841826159, 6.719881205157511, 7.289544571077136, 7.616898883970156, 8.027641022944459, 8.752546163450167, 9.463161614696793, 9.714960095797038, 10.46516902470325, 10.92029381615280, 11.33896493803202, 11.89892141755613, 12.33057500256363, 12.70327014564062, 13.41881340116052, 13.52116660766980

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