Properties

Label 2-85800-1.1-c1-0-28
Degree $2$
Conductor $85800$
Sign $1$
Analytic cond. $685.116$
Root an. cond. $26.1747$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s + 11-s + 13-s + 2·19-s + 2·21-s + 2·23-s + 27-s − 8·29-s + 4·31-s + 33-s − 10·37-s + 39-s + 6·41-s + 12·43-s − 8·47-s − 3·49-s + 6·53-s + 2·57-s − 4·59-s + 6·61-s + 2·63-s + 12·67-s + 2·69-s + 8·71-s + 16·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.458·19-s + 0.436·21-s + 0.417·23-s + 0.192·27-s − 1.48·29-s + 0.718·31-s + 0.174·33-s − 1.64·37-s + 0.160·39-s + 0.937·41-s + 1.82·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.264·57-s − 0.520·59-s + 0.768·61-s + 0.251·63-s + 1.46·67-s + 0.240·69-s + 0.949·71-s + 1.87·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85800\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(685.116\)
Root analytic conductor: \(26.1747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{85800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.142942280\)
\(L(\frac12)\) \(\approx\) \(4.142942280\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
11 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.82924927588342, −13.72302260619043, −12.77449396485486, −12.63698843243537, −11.94853351225961, −11.29967087928321, −11.04659368743094, −10.53575831030986, −9.651353709227493, −9.533998723892202, −8.842196942879995, −8.369159130708954, −7.897478011357540, −7.394935081009441, −6.861050885460433, −6.291257710780898, −5.514644085083569, −5.126748715410762, −4.488274571805098, −3.743743621168928, −3.499701366724244, −2.553376144200735, −2.049900407541030, −1.351381533323151, −0.6589624439086058, 0.6589624439086058, 1.351381533323151, 2.049900407541030, 2.553376144200735, 3.499701366724244, 3.743743621168928, 4.488274571805098, 5.126748715410762, 5.514644085083569, 6.291257710780898, 6.861050885460433, 7.394935081009441, 7.897478011357540, 8.369159130708954, 8.842196942879995, 9.533998723892202, 9.651353709227493, 10.53575831030986, 11.04659368743094, 11.29967087928321, 11.94853351225961, 12.63698843243537, 12.77449396485486, 13.72302260619043, 13.82924927588342

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