Properties

Label 2-185130-1.1-c1-0-123
Degree $2$
Conductor $185130$
Sign $-1$
Analytic cond. $1478.27$
Root an. cond. $38.4482$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 2·14-s + 16-s + 17-s + 20-s + 4·23-s + 25-s + 2·28-s + 2·29-s + 4·31-s − 32-s − 34-s + 2·35-s − 2·37-s − 40-s + 6·43-s − 4·46-s − 3·49-s − 50-s + 6·53-s − 2·56-s − 2·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.377·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s − 0.328·37-s − 0.158·40-s + 0.914·43-s − 0.589·46-s − 3/7·49-s − 0.141·50-s + 0.824·53-s − 0.267·56-s − 0.262·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1478.27\)
Root analytic conductor: \(38.4482\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{185130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 185130,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.37599290377366, −12.90012116723018, −12.26428539522326, −11.96343207836127, −11.35160039931334, −10.98482713117657, −10.47172274619837, −10.13700021199591, −9.508495033930662, −9.088715661811337, −8.688214278313687, −8.050005573035561, −7.800479027234388, −7.147535770683359, −6.657771108595105, −6.216442765909467, −5.485071067460911, −5.194588835360795, −4.497209669450160, −3.993031324831889, −3.114082418504866, −2.706972506052365, −2.059177784380155, −1.358485938246234, −0.9895782058451671, 0, 0.9895782058451671, 1.358485938246234, 2.059177784380155, 2.706972506052365, 3.114082418504866, 3.993031324831889, 4.497209669450160, 5.194588835360795, 5.485071067460911, 6.216442765909467, 6.657771108595105, 7.147535770683359, 7.800479027234388, 8.050005573035561, 8.688214278313687, 9.088715661811337, 9.508495033930662, 10.13700021199591, 10.47172274619837, 10.98482713117657, 11.35160039931334, 11.96343207836127, 12.26428539522326, 12.90012116723018, 13.37599290377366

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