Properties

Label 2-338130-1.1-c1-0-28
Degree $2$
Conductor $338130$
Sign $1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 13-s − 4·14-s + 16-s − 4·19-s − 20-s + 6·23-s + 25-s − 26-s + 4·28-s − 6·29-s + 10·31-s − 32-s − 4·35-s + 10·37-s + 4·38-s + 40-s − 6·41-s − 4·43-s − 6·46-s − 12·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 1.11·29-s + 1.79·31-s − 0.176·32-s − 0.676·35-s + 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.884·46-s − 1.75·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.090110580\)
\(L(\frac12)\) \(\approx\) \(2.090110580\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.71249617903290, −11.78143113732152, −11.52876138828428, −11.16702599253505, −11.01552029812973, −10.30410579141158, −9.759722768262504, −9.510828826600156, −8.629218193062347, −8.377756538908056, −8.193844272595992, −7.669308946093110, −7.112213465812387, −6.550051518110642, −6.315482547286066, −5.346463077853024, −5.109883996784280, −4.547860045157504, −4.041344016218470, −3.441735385892960, −2.742625322776204, −2.238758590364277, −1.585029083487440, −1.125794367134388, −0.4610168907306440, 0.4610168907306440, 1.125794367134388, 1.585029083487440, 2.238758590364277, 2.742625322776204, 3.441735385892960, 4.041344016218470, 4.547860045157504, 5.109883996784280, 5.346463077853024, 6.315482547286066, 6.550051518110642, 7.112213465812387, 7.669308946093110, 8.193844272595992, 8.377756538908056, 8.629218193062347, 9.510828826600156, 9.759722768262504, 10.30410579141158, 11.01552029812973, 11.16702599253505, 11.52876138828428, 11.78143113732152, 12.71249617903290

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