↑

Properties

Label 2-1450-1.1-c1-0-39
Degree $2$
Conductor $1450$
Sign $-1$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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 + 4-s + 8-s − 3·9-s − 2·11-s − 4·13-s + 16-s − 6·17-s − 3·18-s − 8·19-s − 2·22-s + 2·23-s − 4·26-s + 29-s − 31-s + 32-s − 6·34-s − 3·36-s + 7·37-s − 8·38-s + 2·41-s − 4·43-s − 2·44-s + 2·46-s + 9·47-s − 7·49-s − 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.603·11-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 1.83·19-s − 0.426·22-s + 0.417·23-s − 0.784·26-s + 0.185·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.15·37-s − 1.29·38-s + 0.312·41-s − 0.609·43-s − 0.301·44-s + 0.294·46-s + 1.31·47-s − 49-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−8.973234474235627433752292939618, −8.358641250297991841631747538018, −7.36803580465767980930130600607, −6.53889274395399645848113643608, −5.77535928700400652672069823556, −4.83064940807483290068448493427, −4.16681054139743624370465851260, −2.76649614059176809161106912736, −2.23801014953161754739500105156, 0, 2.23801014953161754739500105156, 2.76649614059176809161106912736, 4.16681054139743624370465851260, 4.83064940807483290068448493427, 5.77535928700400652672069823556, 6.53889274395399645848113643608, 7.36803580465767980930130600607, 8.358641250297991841631747538018, 8.973234474235627433752292939618

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