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Properties

Label 2-5400-1.1-c1-0-66
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 − 13-s − 4·17-s − 19-s + 4·23-s − 4·31-s + 9·37-s + 8·43-s − 12·47-s + 2·49-s − 8·53-s − 4·59-s − 5·61-s − 11·67-s − 8·71-s − 73-s − 12·77-s − 5·79-s + 8·83-s − 12·89-s − 3·91-s − 5·97-s − 103-s + 12·107-s − 14·109-s + 12·113-s + ⋯
L(s)  = 1  + 1.13·7-s − 1.20·11-s − 0.277·13-s − 0.970·17-s − 0.229·19-s + 0.834·23-s − 0.718·31-s + 1.47·37-s + 1.21·43-s − 1.75·47-s + 2/7·49-s − 1.09·53-s − 0.520·59-s − 0.640·61-s − 1.34·67-s − 0.949·71-s − 0.117·73-s − 1.36·77-s − 0.562·79-s + 0.878·83-s − 1.27·89-s − 0.314·91-s − 0.507·97-s − 0.0985·103-s + 1.16·107-s − 1.34·109-s + 1.12·113-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 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−7.77958278147023401082479233510, −7.32880674225760895137373999597, −6.34815219966783404362131946963, −5.56804689908969216468713736925, −4.74870427979175472641137796513, −4.43754432829579660950419463063, −3.11229810208127265096597909504, −2.35764756210586239805186250801, −1.44482252057671387648446423733, 0, 1.44482252057671387648446423733, 2.35764756210586239805186250801, 3.11229810208127265096597909504, 4.43754432829579660950419463063, 4.74870427979175472641137796513, 5.56804689908969216468713736925, 6.34815219966783404362131946963, 7.32880674225760895137373999597, 7.77958278147023401082479233510

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