Properties

Label 2-5400-1.1-c1-0-1
Degree $2$
Conductor $5400$
Sign $1$
Analytic cond. $43.1192$
Root an. cond. $6.56652$
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  − 3·7-s − 5·11-s − 4·13-s − 8·17-s + 2·19-s + 2·23-s − 6·29-s − 7·31-s + 6·37-s + 6·41-s + 2·43-s + 6·47-s + 2·49-s + 5·53-s + 4·59-s − 8·61-s + 10·67-s + 8·71-s − 73-s + 15·77-s + 16·79-s − 11·83-s − 6·89-s + 12·91-s + 97-s − 9·101-s − 4·103-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.50·11-s − 1.10·13-s − 1.94·17-s + 0.458·19-s + 0.417·23-s − 1.11·29-s − 1.25·31-s + 0.986·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.686·53-s + 0.520·59-s − 1.02·61-s + 1.22·67-s + 0.949·71-s − 0.117·73-s + 1.70·77-s + 1.80·79-s − 1.20·83-s − 0.635·89-s + 1.25·91-s + 0.101·97-s − 0.895·101-s − 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5400\)    =    \(2^{3} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(43.1192\)
Root analytic conductor: \(6.56652\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6212903381\)
\(L(\frac12)\) \(\approx\) \(0.6212903381\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.069269421184808740710970697688, −7.33268776082441586263181416431, −6.92125321323708344200732959335, −5.96594090077212321671272338363, −5.31975168575974286863951852386, −4.54769035050128794925612182778, −3.65487943004336719014507479967, −2.63349223909005619441201203248, −2.24627128455249706881676401018, −0.38851179148025330850270876693, 0.38851179148025330850270876693, 2.24627128455249706881676401018, 2.63349223909005619441201203248, 3.65487943004336719014507479967, 4.54769035050128794925612182778, 5.31975168575974286863951852386, 5.96594090077212321671272338363, 6.92125321323708344200732959335, 7.33268776082441586263181416431, 8.069269421184808740710970697688

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