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Properties

Label 2-227430-1.1-c1-0-18
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 + 4·11-s + 13-s + 14-s + 16-s − 2·17-s − 20-s − 4·22-s − 6·23-s + 25-s − 26-s − 28-s + 8·29-s + 5·31-s − 32-s + 2·34-s + 35-s − 2·37-s + 40-s − 3·41-s − 7·43-s + 4·44-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 + 1.20·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s − 0.852·22-s − 1.25·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s + 1.48·29-s + 0.898·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.468·41-s − 1.06·43-s + 0.603·44-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

\(L(1)\) \(\approx\) \(0.8440137567\)
\(L(\frac12)\) \(\approx\) \(0.8440137567\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−12.88841873923732, −12.24616335862526, −11.83164434925229, −11.68358362742414, −11.15826837546253, −10.43722680499559, −10.04692607098237, −9.835655835752440, −9.071946528687768, −8.697835386705819, −8.297191400619553, −7.937026472214572, −7.124714151172273, −6.809475170509896, −6.323766619727808, −6.043633441831569, −5.185436372803000, −4.580513978277910, −4.070142177333700, −3.556721349808707, −2.994286685724959, −2.380562854095885, −1.594192972010072, −1.173105632878692, −0.3040677377559374, 0.3040677377559374, 1.173105632878692, 1.594192972010072, 2.380562854095885, 2.994286685724959, 3.556721349808707, 4.070142177333700, 4.580513978277910, 5.185436372803000, 6.043633441831569, 6.323766619727808, 6.809475170509896, 7.124714151172273, 7.937026472214572, 8.297191400619553, 8.697835386705819, 9.071946528687768, 9.835655835752440, 10.04692607098237, 10.43722680499559, 11.15826837546253, 11.68358362742414, 11.83164434925229, 12.24616335862526, 12.88841873923732

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