Properties

Label 2-1950-1.1-c1-0-19
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 + 8-s + 9-s + 12-s + 13-s + 16-s + 6·17-s + 18-s + 4·23-s + 24-s + 26-s + 27-s − 10·29-s + 32-s + 6·34-s + 36-s + 6·37-s + 39-s + 2·41-s + 4·43-s + 4·46-s + 48-s − 7·49-s + 6·51-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.85·29-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.589·46-s + 0.144·48-s − 49-s + 0.840·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.642554850\)
\(L(\frac12)\) \(\approx\) \(3.642554850\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−9.293252484969654106304049481823, −8.234879489422851945430306826139, −7.60700872158310494193298642929, −6.86438211519676053988082818155, −5.82962581845731117670630851923, −5.19858177043401418936672862873, −4.07503419069323835222144097411, −3.40427025338568566249547589245, −2.47214823780223903426583871299, −1.25086079084129802130227088861, 1.25086079084129802130227088861, 2.47214823780223903426583871299, 3.40427025338568566249547589245, 4.07503419069323835222144097411, 5.19858177043401418936672862873, 5.82962581845731117670630851923, 6.86438211519676053988082818155, 7.60700872158310494193298642929, 8.234879489422851945430306826139, 9.293252484969654106304049481823

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