Properties

Label 2-224400-1.1-c1-0-181
Degree $2$
Conductor $224400$
Sign $-1$
Analytic cond. $1791.84$
Root an. cond. $42.3301$
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 − 4·7-s + 9-s + 11-s + 2·13-s − 17-s + 4·19-s − 4·21-s + 27-s + 6·29-s − 4·31-s + 33-s + 6·37-s + 2·39-s + 10·41-s − 4·43-s + 9·49-s − 51-s + 14·53-s + 4·57-s − 14·61-s − 4·63-s − 4·67-s − 10·73-s − 4·77-s + 81-s + 8·83-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 0.872·21-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 9/7·49-s − 0.140·51-s + 1.92·53-s + 0.529·57-s − 1.79·61-s − 0.503·63-s − 0.488·67-s − 1.17·73-s − 0.455·77-s + 1/9·81-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(1791.84\)
Root analytic conductor: \(42.3301\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 224400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.25077511251375, −12.75836689064800, −12.39367330416784, −11.82551049664209, −11.40984387115959, −10.74591186677065, −10.24459580307995, −9.940398903864208, −9.316267216774027, −9.024011450806532, −8.711804965250652, −7.887343991617525, −7.526866285849549, −7.010292651384168, −6.479521793308938, −6.049432866285350, −5.670318271582315, −4.825948646444256, −4.313215338132131, −3.658024501081438, −3.393545497268431, −2.694902161445304, −2.401276901871783, −1.363601130985830, −0.8852935504814922, 0, 0.8852935504814922, 1.363601130985830, 2.401276901871783, 2.694902161445304, 3.393545497268431, 3.658024501081438, 4.313215338132131, 4.825948646444256, 5.670318271582315, 6.049432866285350, 6.479521793308938, 7.010292651384168, 7.526866285849549, 7.887343991617525, 8.711804965250652, 9.024011450806532, 9.316267216774027, 9.940398903864208, 10.24459580307995, 10.74591186677065, 11.40984387115959, 11.82551049664209, 12.39367330416784, 12.75836689064800, 13.25077511251375

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