Properties

Label 2-101430-1.1-c1-0-97
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 2·11-s + 6·13-s + 16-s − 6·17-s + 20-s + 2·22-s − 23-s + 25-s − 6·26-s + 4·29-s + 2·31-s − 32-s + 6·34-s + 4·37-s − 40-s − 2·41-s − 4·43-s − 2·44-s + 46-s − 50-s + 6·52-s − 6·53-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 + 1.66·13-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 0.426·22-s − 0.208·23-s + 1/5·25-s − 1.17·26-s + 0.742·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s + 0.657·37-s − 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.301·44-s + 0.147·46-s − 0.141·50-s + 0.832·52-s − 0.824·53-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: $\chi_{101430} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 101430,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.87403116555417, −13.49878781685054, −13.00198893096136, −12.67143618395979, −11.86161174142266, −11.28238894776742, −11.11433849018406, −10.44123960194217, −10.14797420724067, −9.500194152598711, −8.934436040121338, −8.588462341111469, −8.131275961245666, −7.626660799685152, −6.756536802020286, −6.532546161248787, −6.011934604461606, −5.444178806226057, −4.703606856369216, −4.186716786677585, −3.413242556815916, −2.827443135552091, −2.181815106964159, −1.569752630263900, −0.8876920963523582, 0, 0.8876920963523582, 1.569752630263900, 2.181815106964159, 2.827443135552091, 3.413242556815916, 4.186716786677585, 4.703606856369216, 5.444178806226057, 6.011934604461606, 6.532546161248787, 6.756536802020286, 7.626660799685152, 8.131275961245666, 8.588462341111469, 8.934436040121338, 9.500194152598711, 10.14797420724067, 10.44123960194217, 11.11433849018406, 11.28238894776742, 11.86161174142266, 12.67143618395979, 13.00198893096136, 13.49878781685054, 13.87403116555417

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