Properties

Label 2-117600-1.1-c1-0-121
Degree $2$
Conductor $117600$
Sign $-1$
Analytic cond. $939.040$
Root an. cond. $30.6437$
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-s + 9-s + 3·11-s + 13-s − 4·17-s + 19-s − 5·23-s − 27-s + 4·29-s − 2·31-s − 3·33-s + 3·37-s − 39-s − 3·41-s + 4·43-s − 9·47-s + 4·51-s − 3·53-s − 57-s + 12·59-s − 4·67-s + 5·69-s − 10·71-s + 2·73-s + 81-s + 8·83-s − 4·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.970·17-s + 0.229·19-s − 1.04·23-s − 0.192·27-s + 0.742·29-s − 0.359·31-s − 0.522·33-s + 0.493·37-s − 0.160·39-s − 0.468·41-s + 0.609·43-s − 1.31·47-s + 0.560·51-s − 0.412·53-s − 0.132·57-s + 1.56·59-s − 0.488·67-s + 0.601·69-s − 1.18·71-s + 0.234·73-s + 1/9·81-s + 0.878·83-s − 0.428·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117600\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(939.040\)
Root analytic conductor: \(30.6437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{117600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 117600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.81883874228805, −13.28255033568370, −12.89762549936853, −12.25911420158841, −11.85716805656031, −11.41784916732117, −11.07889720105482, −10.37658403590505, −10.02958401787697, −9.449369406188386, −8.924570768226757, −8.499029568908559, −7.850136698193558, −7.349570553365510, −6.614053498796501, −6.414744767112616, −5.896543266162414, −5.201639500637838, −4.676714590180890, −4.098414980558068, −3.687327592437887, −2.907959933596193, −2.136283668981326, −1.560417829729421, −0.8284799084388063, 0, 0.8284799084388063, 1.560417829729421, 2.136283668981326, 2.907959933596193, 3.687327592437887, 4.098414980558068, 4.676714590180890, 5.201639500637838, 5.896543266162414, 6.414744767112616, 6.614053498796501, 7.349570553365510, 7.850136698193558, 8.499029568908559, 8.924570768226757, 9.449369406188386, 10.02958401787697, 10.37658403590505, 11.07889720105482, 11.41784916732117, 11.85716805656031, 12.25911420158841, 12.89762549936853, 13.28255033568370, 13.81883874228805

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