↑

Properties

Label 2-379050-1.1-c1-0-30
Degree $2$
Conductor $379050$
Sign $1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 6·11-s − 12-s − 2·13-s − 14-s + 16-s − 2·17-s − 18-s − 21-s + 6·22-s + 24-s + 2·26-s − 27-s + 28-s − 2·29-s + 6·31-s − 32-s + 6·33-s + 2·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.218·21-s + 1.27·22-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 1.07·31-s − 0.176·32-s + 1.04·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9242622529\)
\(L(\frac12)\) \(\approx\) \(0.9242622529\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
19 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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

−12.55392504315255, −11.76657711570606, −11.59480997569492, −11.04507582897343, −10.65431887434579, −10.18553065736407, −9.910703987271905, −9.472650979971393, −8.700084420183761, −8.345623908681091, −7.949503503438502, −7.538130902946585, −6.926914167594881, −6.677723019030622, −5.963679221366898, −5.470996917088144, −4.981018091279987, −4.784157385178044, −3.950063615132161, −3.356236333343096, −2.592049489357051, −2.303455491342446, −1.734341999995324, −0.8294958666703334, −0.3657782251687786, 0.3657782251687786, 0.8294958666703334, 1.734341999995324, 2.303455491342446, 2.592049489357051, 3.356236333343096, 3.950063615132161, 4.784157385178044, 4.981018091279987, 5.470996917088144, 5.963679221366898, 6.677723019030622, 6.926914167594881, 7.538130902946585, 7.949503503438502, 8.345623908681091, 8.700084420183761, 9.472650979971393, 9.910703987271905, 10.18553065736407, 10.65431887434579, 11.04507582897343, 11.59480997569492, 11.76657711570606, 12.55392504315255

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