Properties

Label 2-273600-1.1-c1-0-261
Degree $2$
Conductor $273600$
Sign $-1$
Analytic cond. $2184.70$
Root an. cond. $46.7408$
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  − 4·11-s + 2·13-s + 2·17-s − 19-s − 4·23-s + 6·29-s − 4·31-s − 6·37-s − 10·41-s + 4·43-s + 12·47-s − 7·49-s − 6·53-s + 12·59-s + 2·61-s − 4·67-s + 8·71-s + 6·73-s + 4·79-s − 12·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.229·19-s − 0.834·23-s + 1.11·29-s − 0.718·31-s − 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.75·47-s − 49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.450·79-s − 1.31·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(2184.70\)
Root analytic conductor: \(46.7408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{273600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 273600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.01291574593316, −12.51622937035768, −12.12748936290825, −11.70570833042958, −10.99577107785509, −10.75558120523419, −10.20845628312688, −9.921466267162091, −9.348453170054466, −8.669963018053067, −8.315251017063653, −8.019690674808508, −7.350248066682378, −6.953427112182894, −6.368002199590307, −5.859090519147186, −5.283124003889653, −5.070307909568357, −4.262567161903138, −3.809861360046974, −3.225566881808338, −2.691722918248576, −2.085968525446568, −1.515468989305711, −0.7078118080168380, 0, 0.7078118080168380, 1.515468989305711, 2.085968525446568, 2.691722918248576, 3.225566881808338, 3.809861360046974, 4.262567161903138, 5.070307909568357, 5.283124003889653, 5.859090519147186, 6.368002199590307, 6.953427112182894, 7.350248066682378, 8.019690674808508, 8.315251017063653, 8.669963018053067, 9.348453170054466, 9.921466267162091, 10.20845628312688, 10.75558120523419, 10.99577107785509, 11.70570833042958, 12.12748936290825, 12.51622937035768, 13.01291574593316

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