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Properties

Label 2-227430-1.1-c1-0-79
Degree $2$
Conductor $227430$
Sign $1$
Analytic cond. $1816.03$
Root an. cond. $42.6149$
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 − 7-s − 8-s + 10-s − 3·11-s − 13-s + 14-s + 16-s + 4·17-s − 20-s + 3·22-s + 3·23-s + 25-s + 26-s − 28-s + 8·29-s − 3·31-s − 32-s − 4·34-s + 35-s + 4·37-s + 40-s + 11·43-s − 3·44-s − 3·46-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 − 0.904·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 0.639·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.48·29-s − 0.538·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s + 0.657·37-s + 0.158·40-s + 1.67·43-s − 0.452·44-s − 0.442·46-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: $1$
Analytic conductor: \(1816.03\)
Root analytic conductor: \(42.6149\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.922297019\)
\(L(\frac12)\) \(\approx\) \(1.922297019\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.72073884075070, −12.42702990637360, −12.07716371006699, −11.52156425219229, −10.88564305373699, −10.53340062852197, −10.31178943267475, −9.544931687185219, −9.260171113040806, −8.757412306816822, −8.150252391908700, −7.713362208585611, −7.412615475388053, −6.957985498336441, −6.091509354170928, −6.004188490326677, −5.155566924835205, −4.769510812269306, −4.093126660578901, −3.409322243368778, −2.911489334936459, −2.499588501954268, −1.772127713812696, −0.7360860998776830, −0.6551134949593839, 0.6551134949593839, 0.7360860998776830, 1.772127713812696, 2.499588501954268, 2.911489334936459, 3.409322243368778, 4.093126660578901, 4.769510812269306, 5.155566924835205, 6.004188490326677, 6.091509354170928, 6.957985498336441, 7.412615475388053, 7.713362208585611, 8.150252391908700, 8.757412306816822, 9.260171113040806, 9.544931687185219, 10.31178943267475, 10.53340062852197, 10.88564305373699, 11.52156425219229, 12.07716371006699, 12.42702990637360, 12.72073884075070

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