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Properties

Label 2-10800-1.1-c1-0-36
Degree $2$
Conductor $10800$
Sign $1$
Analytic cond. $86.2384$
Root an. cond. $9.28646$
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  + 4·7-s + 4·11-s + 13-s − 8·17-s + 8·19-s − 8·23-s − 4·29-s − 31-s + 3·37-s + 11·43-s − 8·47-s + 9·49-s + 12·53-s + 8·59-s − 2·61-s − 11·67-s + 12·71-s + 9·73-s + 16·77-s − 9·79-s − 4·83-s + 12·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s + 0.277·13-s − 1.94·17-s + 1.83·19-s − 1.66·23-s − 0.742·29-s − 0.179·31-s + 0.493·37-s + 1.67·43-s − 1.16·47-s + 9/7·49-s + 1.64·53-s + 1.04·59-s − 0.256·61-s − 1.34·67-s + 1.42·71-s + 1.05·73-s + 1.82·77-s − 1.01·79-s − 0.439·83-s + 1.27·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(86.2384\)
Root analytic conductor: \(9.28646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.859096424\)
\(L(\frac12)\) \(\approx\) \(2.859096424\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.50398786750529, −15.92473663375741, −15.40887434393158, −14.66729923033156, −14.30831652096610, −13.73751912138314, −13.30187043693568, −12.31443403905366, −11.75670074190375, −11.32004196299050, −11.01414854146843, −10.05366554108798, −9.393318753252353, −8.837290379118524, −8.274231786413014, −7.554001930612427, −7.056650061004026, −6.169470839863158, −5.605567604472876, −4.743718014444039, −4.201586220257976, −3.587738346204971, −2.300388090461680, −1.750672004769429, −0.8239395043608689, 0.8239395043608689, 1.750672004769429, 2.300388090461680, 3.587738346204971, 4.201586220257976, 4.743718014444039, 5.605567604472876, 6.169470839863158, 7.056650061004026, 7.554001930612427, 8.274231786413014, 8.837290379118524, 9.393318753252353, 10.05366554108798, 11.01414854146843, 11.32004196299050, 11.75670074190375, 12.31443403905366, 13.30187043693568, 13.73751912138314, 14.30831652096610, 14.66729923033156, 15.40887434393158, 15.92473663375741, 16.50398786750529

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