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Properties

Label 2-227430-1.1-c1-0-0
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 − 2·11-s − 2·13-s + 14-s + 16-s − 5·17-s − 20-s + 2·22-s − 6·23-s + 25-s + 2·26-s − 28-s − 4·29-s − 4·31-s − 32-s + 5·34-s + 35-s + 7·37-s + 40-s + 6·41-s − 13·43-s − 2·44-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 − 0.603·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.223·20-s + 0.426·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.742·29-s − 0.718·31-s − 0.176·32-s + 0.857·34-s + 0.169·35-s + 1.15·37-s + 0.158·40-s + 0.937·41-s − 1.98·43-s − 0.301·44-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

\(L(1)\) \(\approx\) \(0.01793509853\)
\(L(\frac12)\) \(\approx\) \(0.01793509853\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−12.83688302808556, −12.42562142815443, −12.00547292071546, −11.41377808937340, −11.03288810790794, −10.65028633426922, −10.11564299371907, −9.541695373602404, −9.324234348060283, −8.734405576463590, −8.104951350467634, −7.864566947303641, −7.324413558250845, −6.861382268597463, −6.311269767953286, −5.855240831184252, −5.262560802987723, −4.581611484425804, −4.168979683392206, −3.482429320474748, −2.958243427198304, −2.185095730975227, −2.004032364135674, −0.9877845624157083, −0.04613603826418059, 0.04613603826418059, 0.9877845624157083, 2.004032364135674, 2.185095730975227, 2.958243427198304, 3.482429320474748, 4.168979683392206, 4.581611484425804, 5.262560802987723, 5.855240831184252, 6.311269767953286, 6.861382268597463, 7.324413558250845, 7.864566947303641, 8.104951350467634, 8.734405576463590, 9.324234348060283, 9.541695373602404, 10.11564299371907, 10.65028633426922, 11.03288810790794, 11.41377808937340, 12.00547292071546, 12.42562142815443, 12.83688302808556

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