Properties

Label 2-117600-1.1-c1-0-49
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·11-s + 6·13-s − 6·17-s + 4·19-s − 8·23-s − 27-s + 10·29-s + 4·31-s + 4·33-s + 6·37-s − 6·39-s − 6·41-s + 4·43-s + 12·47-s + 6·51-s − 6·53-s − 4·57-s + 4·59-s + 2·61-s + 4·67-s + 8·69-s − 2·73-s + 8·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s + 0.986·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.963·69-s − 0.234·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-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: \(0\)
Selberg data: \((2,\ 117600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934530729\)
\(L(\frac12)\) \(\approx\) \(1.934530729\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.55330186392606, −13.19112760211629, −12.64609798697480, −11.99502658603166, −11.72244455993245, −11.08686562692144, −10.70489942461262, −10.30339020629031, −9.834546229501197, −9.141935976128729, −8.617474229941711, −8.036967333840397, −7.867363700307072, −6.941543956774673, −6.527313935390201, −6.019047659378486, −5.631884833984712, −4.941085462658048, −4.393623621527073, −3.960082281066867, −3.187267088723402, −2.538290894256664, −1.971501226669912, −1.052451844894350, −0.5108284773752048, 0.5108284773752048, 1.052451844894350, 1.971501226669912, 2.538290894256664, 3.187267088723402, 3.960082281066867, 4.393623621527073, 4.941085462658048, 5.631884833984712, 6.019047659378486, 6.527313935390201, 6.941543956774673, 7.867363700307072, 8.036967333840397, 8.617474229941711, 9.141935976128729, 9.834546229501197, 10.30339020629031, 10.70489942461262, 11.08686562692144, 11.72244455993245, 11.99502658603166, 12.64609798697480, 13.19112760211629, 13.55330186392606

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