↑

Properties

Label 2-48510-1.1-c1-0-64
Degree $2$
Conductor $48510$
Sign $-1$
Analytic cond. $387.354$
Root an. cond. $19.6813$
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 − 5-s − 8-s + 10-s − 11-s + 2·13-s + 16-s + 6·17-s − 8·19-s − 20-s + 22-s + 25-s − 2·26-s + 2·29-s + 4·31-s − 32-s − 6·34-s − 2·37-s + 8·38-s + 40-s − 2·41-s + 4·43-s − 44-s − 4·47-s − 50-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s − 1.83·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s + 1.29·38-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.150·44-s − 0.583·47-s − 0.141·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48510\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(387.354\)
Root analytic conductor: \(19.6813\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 48510,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.94996482456733, −14.30657499552187, −13.96513658205978, −13.10765455459566, −12.66554824999679, −12.23382873414572, −11.68633636191267, −11.05557176880925, −10.62566223376544, −10.24190776120136, −9.572578744262840, −9.068892683387183, −8.324659203861003, −8.097458604264253, −7.643779632127393, −6.793535798094235, −6.396033073164474, −5.842881214066330, −5.053692386368250, −4.463748852105188, −3.681875657109095, −3.174202258726804, −2.406511443098234, −1.644385717224528, −0.8775283337252859, 0, 0.8775283337252859, 1.644385717224528, 2.406511443098234, 3.174202258726804, 3.681875657109095, 4.463748852105188, 5.053692386368250, 5.842881214066330, 6.396033073164474, 6.793535798094235, 7.643779632127393, 8.097458604264253, 8.324659203861003, 9.068892683387183, 9.572578744262840, 10.24190776120136, 10.62566223376544, 11.05557176880925, 11.68633636191267, 12.23382873414572, 12.66554824999679, 13.10765455459566, 13.96513658205978, 14.30657499552187, 14.94996482456733

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