Properties

Label 2-5400-1.1-c1-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s − 4·11-s + 6·13-s − 6·17-s − 19-s + 6·23-s + 10·29-s + 31-s + 11·37-s − 3·43-s − 8·47-s + 2·49-s − 12·53-s + 6·59-s + 5·61-s − 4·67-s + 2·71-s − 9·73-s + 12·77-s − 5·79-s − 18·83-s − 2·89-s − 18·91-s − 5·97-s − 9·103-s − 2·107-s − 9·109-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.229·19-s + 1.25·23-s + 1.85·29-s + 0.179·31-s + 1.80·37-s − 0.457·43-s − 1.16·47-s + 2/7·49-s − 1.64·53-s + 0.781·59-s + 0.640·61-s − 0.488·67-s + 0.237·71-s − 1.05·73-s + 1.36·77-s − 0.562·79-s − 1.97·83-s − 0.211·89-s − 1.88·91-s − 0.507·97-s − 0.886·103-s − 0.193·107-s − 0.862·109-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.010094141709394340009585013996, −6.78797813309235463835122476926, −6.53760587112111112704725722414, −5.79689204333495255763372859037, −4.82943887504030152922335792770, −4.13160278385362758222196589746, −3.05744413614205800801394380968, −2.66927901345454467416597086485, −1.24035155380595107614220150390, 0, 1.24035155380595107614220150390, 2.66927901345454467416597086485, 3.05744413614205800801394380968, 4.13160278385362758222196589746, 4.82943887504030152922335792770, 5.79689204333495255763372859037, 6.53760587112111112704725722414, 6.78797813309235463835122476926, 8.010094141709394340009585013996

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