Properties

Label 2-420e2-1.1-c1-0-495
Degree $2$
Conductor $176400$
Sign $1$
Analytic cond. $1408.56$
Root an. cond. $37.5308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·13-s − 2·17-s − 8·19-s − 8·23-s + 2·29-s + 4·31-s + 2·37-s − 6·41-s + 4·43-s + 8·47-s + 10·53-s − 4·59-s + 2·61-s + 4·67-s − 12·71-s − 2·73-s − 8·79-s − 4·83-s − 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.66·13-s − 0.485·17-s − 1.83·19-s − 1.66·23-s + 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.488·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1408.56\)
Root analytic conductor: \(37.5308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{176400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

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

−13.67346843961417, −13.17693758324475, −12.58153813909968, −12.24837390209479, −11.91363096314277, −11.34861732103148, −10.71667645902272, −10.24475309912066, −10.00743338222712, −9.450801990624473, −8.747374463524334, −8.488519166928601, −7.880361096255469, −7.411374929703662, −6.836598055696052, −6.439033325130713, −5.845471756201932, −5.346325400956165, −4.659968196458529, −4.156372566304164, −3.983171754618743, −2.819095466228097, −2.479014035366735, −2.036028346215775, −1.214946165904799, 0, 0, 1.214946165904799, 2.036028346215775, 2.479014035366735, 2.819095466228097, 3.983171754618743, 4.156372566304164, 4.659968196458529, 5.346325400956165, 5.845471756201932, 6.439033325130713, 6.836598055696052, 7.411374929703662, 7.880361096255469, 8.488519166928601, 8.747374463524334, 9.450801990624473, 10.00743338222712, 10.24475309912066, 10.71667645902272, 11.34861732103148, 11.91363096314277, 12.24837390209479, 12.58153813909968, 13.17693758324475, 13.67346843961417

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