Properties

Label 2-2070-1.1-c1-0-20
Degree $2$
Conductor $2070$
Sign $1$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 2·11-s + 4·14-s + 16-s − 2·17-s + 20-s + 2·22-s − 23-s + 25-s + 4·28-s + 4·29-s + 32-s − 2·34-s + 4·35-s + 10·37-s + 40-s − 6·41-s + 2·43-s + 2·44-s − 46-s − 12·47-s + 9·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.426·22-s − 0.208·23-s + 1/5·25-s + 0.755·28-s + 0.742·29-s + 0.176·32-s − 0.342·34-s + 0.676·35-s + 1.64·37-s + 0.158·40-s − 0.937·41-s + 0.304·43-s + 0.301·44-s − 0.147·46-s − 1.75·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.739002811\)
\(L(\frac12)\) \(\approx\) \(3.739002811\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−9.076091713653181010715658475879, −8.208781540882969526831297540215, −7.60416359625391549446622435642, −6.54339823395422894886744855741, −5.96242520137679473951679895257, −4.79077329145906956219957278889, −4.59738734063357382886923854131, −3.33419874663372898406963550539, −2.16617393996501176803226728421, −1.34409171825676144708222230132, 1.34409171825676144708222230132, 2.16617393996501176803226728421, 3.33419874663372898406963550539, 4.59738734063357382886923854131, 4.79077329145906956219957278889, 5.96242520137679473951679895257, 6.54339823395422894886744855741, 7.60416359625391549446622435642, 8.208781540882969526831297540215, 9.076091713653181010715658475879

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