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Properties

Label 2-101430-1.1-c1-0-36
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 + 4·11-s + 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s − 4·22-s + 23-s + 25-s − 2·26-s − 2·29-s − 4·31-s − 32-s − 6·34-s − 6·37-s − 4·38-s + 40-s + 10·41-s + 4·43-s + 4·44-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 + 1.20·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.986·37-s − 0.648·38-s + 0.158·40-s + 1.56·41-s + 0.609·43-s + 0.603·44-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

\(L(1)\) \(\approx\) \(2.036217820\)
\(L(\frac12)\) \(\approx\) \(2.036217820\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.93162932677023, −13.10379556077776, −12.69892433760097, −12.10848450315127, −11.69068323473766, −11.35810694287795, −10.82222311634626, −10.23137947883786, −9.675878102683762, −9.349437007775974, −8.753808717738787, −8.353178311974988, −7.681300958837512, −7.310052546161524, −6.866516397011782, −6.145922088701930, −5.686218798519551, −5.157285266893387, −4.295884306685512, −3.683220506463055, −3.348032569531419, −2.644978277644576, −1.633370397685812, −1.243202588622378, −0.5584991928536916, 0.5584991928536916, 1.243202588622378, 1.633370397685812, 2.644978277644576, 3.348032569531419, 3.683220506463055, 4.295884306685512, 5.157285266893387, 5.686218798519551, 6.145922088701930, 6.866516397011782, 7.310052546161524, 7.681300958837512, 8.353178311974988, 8.753808717738787, 9.349437007775974, 9.675878102683762, 10.23137947883786, 10.82222311634626, 11.35810694287795, 11.69068323473766, 12.10848450315127, 12.69892433760097, 13.10379556077776, 13.93162932677023

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