Properties

Degree $2$
Conductor $227430$
Sign $-1$
Motivic weight $1$
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 − 7-s − 8-s + 10-s + 4·11-s − 2·13-s + 14-s + 16-s − 6·17-s − 20-s − 4·22-s + 4·23-s + 25-s + 2·26-s − 28-s + 6·29-s + 4·31-s − 32-s + 6·34-s + 35-s + 6·37-s + 40-s + 6·41-s + 4·44-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.986·37-s + 0.158·40-s + 0.937·41-s + 0.603·44-s − 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(227430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{227430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ -1)\)

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

−13.00434073869135, −12.68228274311530, −12.11319604858062, −11.72135457908912, −11.27295062967600, −10.90934796272684, −10.36551546872461, −9.767483795020395, −9.426356098538082, −8.939297098511038, −8.592205925019785, −8.052672283113531, −7.500555738968890, −6.926100710893465, −6.571898122675885, −6.335887595264607, −5.500716976717875, −4.895248553778585, −4.220574682504381, −4.015010116586189, −3.183016055928506, −2.542313938585630, −2.257821281201166, −1.187790524134548, −0.8437146266131944, 0, 0.8437146266131944, 1.187790524134548, 2.257821281201166, 2.542313938585630, 3.183016055928506, 4.015010116586189, 4.220574682504381, 4.895248553778585, 5.500716976717875, 6.335887595264607, 6.571898122675885, 6.926100710893465, 7.500555738968890, 8.052672283113531, 8.592205925019785, 8.939297098511038, 9.426356098538082, 9.767483795020395, 10.36551546872461, 10.90934796272684, 11.27295062967600, 11.72135457908912, 12.11319604858062, 12.68228274311530, 13.00434073869135

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