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Properties

Label 2-5400-1.1-c1-0-48
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  − 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 − 5·103-s + 12·107-s + 5·109-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 − 0.492·103-s + 1.16·107-s + 0.478·109-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 \( 1 + 4 T + p T^{2} \)
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

−8.037292523181959873472251100651, −6.78914757341740301720263668765, −6.41071484321467520227200318505, −5.96082579429636894372966668019, −4.86507388624601373776077078985, −3.69651000614356009545808864120, −3.61734768315851682332839612613, −2.41609198679500346683242484753, −1.29427729624552858153974947022, 0, 1.29427729624552858153974947022, 2.41609198679500346683242484753, 3.61734768315851682332839612613, 3.69651000614356009545808864120, 4.86507388624601373776077078985, 5.96082579429636894372966668019, 6.41071484321467520227200318505, 6.78914757341740301720263668765, 8.037292523181959873472251100651

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