Properties

Label 2-162288-1.1-c1-0-24
Degree $2$
Conductor $162288$
Sign $1$
Analytic cond. $1295.87$
Root an. cond. $35.9982$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s + 6·17-s − 6·19-s − 23-s − 5·25-s − 10·29-s + 4·31-s − 2·37-s − 10·41-s + 4·43-s − 12·47-s + 6·53-s + 2·59-s − 8·71-s + 6·73-s + 8·79-s + 14·83-s − 14·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 1.45·17-s − 1.37·19-s − 0.208·23-s − 25-s − 1.85·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 0.824·53-s + 0.260·59-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1.53·83-s − 1.48·89-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162288\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1295.87\)
Root analytic conductor: \(35.9982\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162288} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162288,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.687721737\)
\(L(\frac12)\) \(\approx\) \(1.687721737\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.31888622223207, −12.72989684090412, −12.29130090553936, −11.78704306001226, −11.52116748820753, −10.92556773387436, −10.33145564116688, −9.887314360740399, −9.522163950350100, −8.938351619972000, −8.456919482319139, −7.949305054298049, −7.496750533083307, −6.873010845414621, −6.350366448351614, −6.007562436965736, −5.335182958605284, −4.892180448787572, −3.982537121157146, −3.836038887708094, −3.275629773302998, −2.413568312062768, −1.751694465775790, −1.357913775499097, −0.3742548448023080, 0.3742548448023080, 1.357913775499097, 1.751694465775790, 2.413568312062768, 3.275629773302998, 3.836038887708094, 3.982537121157146, 4.892180448787572, 5.335182958605284, 6.007562436965736, 6.350366448351614, 6.873010845414621, 7.496750533083307, 7.949305054298049, 8.456919482319139, 8.938351619972000, 9.522163950350100, 9.887314360740399, 10.33145564116688, 10.92556773387436, 11.52116748820753, 11.78704306001226, 12.29130090553936, 12.72989684090412, 13.31888622223207

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