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Properties

Label 2-117600-1.1-c1-0-124
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 − 2·13-s − 6·17-s − 4·23-s + 27-s − 2·29-s − 8·31-s − 6·37-s − 2·39-s + 6·41-s + 12·43-s + 12·47-s − 6·51-s + 10·53-s + 8·59-s + 10·61-s − 12·67-s − 4·69-s − 8·71-s + 10·73-s − 16·79-s + 81-s − 12·83-s − 2·87-s + 6·89-s − 8·93-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.834·23-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 1.82·43-s + 1.75·47-s − 0.840·51-s + 1.37·53-s + 1.04·59-s + 1.28·61-s − 1.46·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s − 1.80·79-s + 1/9·81-s − 1.31·83-s − 0.214·87-s + 0.635·89-s − 0.829·93-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: Trivial
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.85796493845750, −13.35383592924435, −12.87754960401658, −12.50151544405609, −11.94316898587471, −11.36196405157674, −10.92849937688494, −10.33497157463311, −10.00606526648740, −9.243846885871262, −8.834633288952861, −8.711959343041578, −7.783324868342744, −7.319799440779650, −7.111623622634169, −6.343832712376262, −5.676799554697859, −5.367054896700898, −4.381768579136793, −4.163418294129491, −3.618080181607108, −2.716122876379752, −2.288267541133918, −1.839137474081811, −0.8325870320135962, 0, 0.8325870320135962, 1.839137474081811, 2.288267541133918, 2.716122876379752, 3.618080181607108, 4.163418294129491, 4.381768579136793, 5.367054896700898, 5.676799554697859, 6.343832712376262, 7.111623622634169, 7.319799440779650, 7.783324868342744, 8.711959343041578, 8.834633288952861, 9.243846885871262, 10.00606526648740, 10.33497157463311, 10.92849937688494, 11.36196405157674, 11.94316898587471, 12.50151544405609, 12.87754960401658, 13.35383592924435, 13.85796493845750

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