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Properties

Label 2-227430-1.1-c1-0-1
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 − 5·13-s + 14-s + 16-s − 5·17-s − 20-s − 3·22-s − 7·23-s + 25-s + 5·26-s − 28-s − 3·29-s + 8·31-s − 32-s + 5·34-s + 35-s + 40-s − 5·41-s + 3·44-s + 7·46-s + 49-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 − 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.223·20-s − 0.639·22-s − 1.45·23-s + 1/5·25-s + 0.980·26-s − 0.188·28-s − 0.557·29-s + 1.43·31-s − 0.176·32-s + 0.857·34-s + 0.169·35-s + 0.158·40-s − 0.780·41-s + 0.452·44-s + 1.03·46-s + 1/7·49-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

\(L(1)\) \(\approx\) \(0.1633747179\)
\(L(\frac12)\) \(\approx\) \(0.1633747179\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.78308399953932, −12.31171527274312, −11.95692559682758, −11.58828170002181, −11.11458681019593, −10.52682236561089, −10.04768199348905, −9.609456242806313, −9.315450687024523, −8.728311990119042, −8.148525499566916, −7.916648478531109, −7.191230507442045, −6.781901473243634, −6.474938794487993, −5.882514067091386, −5.196390878771959, −4.564132264789188, −4.139778480924946, −3.597441329038719, −2.860232196816530, −2.342887027672715, −1.828090943566105, −1.050394883429005, −0.1357591614044371, 0.1357591614044371, 1.050394883429005, 1.828090943566105, 2.342887027672715, 2.860232196816530, 3.597441329038719, 4.139778480924946, 4.564132264789188, 5.196390878771959, 5.882514067091386, 6.474938794487993, 6.781901473243634, 7.191230507442045, 7.916648478531109, 8.148525499566916, 8.728311990119042, 9.315450687024523, 9.609456242806313, 10.04768199348905, 10.52682236561089, 11.11458681019593, 11.58828170002181, 11.95692559682758, 12.31171527274312, 12.78308399953932

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