Properties

Label 2-51842-1.1-c1-0-10
Degree $2$
Conductor $51842$
Sign $-1$
Analytic cond. $413.960$
Root an. cond. $20.3460$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 3·5-s − 2·6-s − 8-s + 9-s + 3·10-s + 6·11-s + 2·12-s + 13-s − 6·15-s + 16-s − 6·17-s − 18-s − 2·19-s − 3·20-s − 6·22-s − 2·24-s + 4·25-s − 26-s − 4·27-s − 9·29-s + 6·30-s + 4·31-s − 32-s + 12·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.80·11-s + 0.577·12-s + 0.277·13-s − 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.670·20-s − 1.27·22-s − 0.408·24-s + 4/5·25-s − 0.196·26-s − 0.769·27-s − 1.67·29-s + 1.09·30-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51842\)    =    \(2 \cdot 7^{2} \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(413.960\)
Root analytic conductor: \(20.3460\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51842} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 51842,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−14.86692397723530, −14.39491373551152, −13.85916416091419, −13.16641608755470, −12.75624028099747, −11.94626705188856, −11.58577042354898, −11.17027824659124, −10.79559738624569, −9.760236061327746, −9.375224023945534, −8.920528616867979, −8.498645513557648, −8.126706640474729, −7.418433309756591, −7.105888777087304, −6.364201456476319, −5.967264929824126, −4.687734960274530, −4.215267565637384, −3.632176534094180, −3.330018072277435, −2.315569237628697, −1.860770220471151, −0.9050240765993498, 0, 0.9050240765993498, 1.860770220471151, 2.315569237628697, 3.330018072277435, 3.632176534094180, 4.215267565637384, 4.687734960274530, 5.967264929824126, 6.364201456476319, 7.105888777087304, 7.418433309756591, 8.126706640474729, 8.498645513557648, 8.920528616867979, 9.375224023945534, 9.760236061327746, 10.79559738624569, 11.17027824659124, 11.58577042354898, 11.94626705188856, 12.75624028099747, 13.16641608755470, 13.85916416091419, 14.39491373551152, 14.86692397723530

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