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Properties

Label 2-227430-1.1-c1-0-35
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 − 7·17-s − 20-s + 3·22-s + 7·23-s + 25-s + 26-s + 28-s − 3·29-s + 32-s − 7·34-s − 35-s + 4·37-s − 40-s + 3·41-s − 8·43-s + 3·44-s + 7·46-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 − 1.69·17-s − 0.223·20-s + 0.639·22-s + 1.45·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.557·29-s + 0.176·32-s − 1.20·34-s − 0.169·35-s + 0.657·37-s − 0.158·40-s + 0.468·41-s − 1.21·43-s + 0.452·44-s + 1.03·46-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

\(L(1)\) \(\approx\) \(3.395613793\)
\(L(\frac12)\) \(\approx\) \(3.395613793\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.12559608012960, −12.52223121335455, −11.89819456123746, −11.62127253690746, −11.10209761589681, −10.90780976566230, −10.36298060415889, −9.508865213746224, −9.191727291852898, −8.725444733293338, −8.221519015396707, −7.638427595291167, −7.137410547860747, −6.590500159857591, −6.399293859149934, −5.693393720484203, −5.000352419367601, −4.683836692115614, −4.139018845107482, −3.747748239493668, −3.012933721176244, −2.622661938043860, −1.708110615469879, −1.401501902693330, −0.4393602510022277, 0.4393602510022277, 1.401501902693330, 1.708110615469879, 2.622661938043860, 3.012933721176244, 3.747748239493668, 4.139018845107482, 4.683836692115614, 5.000352419367601, 5.693393720484203, 6.399293859149934, 6.590500159857591, 7.137410547860747, 7.638427595291167, 8.221519015396707, 8.725444733293338, 9.191727291852898, 9.508865213746224, 10.36298060415889, 10.90780976566230, 11.10209761589681, 11.62127253690746, 11.89819456123746, 12.52223121335455, 13.12559608012960

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