Properties

Label 2-177744-1.1-c1-0-57
Degree $2$
Conductor $177744$
Sign $-1$
Analytic cond. $1419.29$
Root an. cond. $37.6735$
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 − 3·5-s − 7-s + 9-s + 4·11-s − 3·13-s + 3·15-s + 4·17-s + 21-s + 4·25-s − 27-s + 3·29-s + 6·31-s − 4·33-s + 3·35-s + 9·37-s + 3·39-s + 9·41-s − 3·43-s − 3·45-s + 7·47-s + 49-s − 4·51-s + 4·53-s − 12·55-s − 6·59-s − 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.832·13-s + 0.774·15-s + 0.970·17-s + 0.218·21-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 1.07·31-s − 0.696·33-s + 0.507·35-s + 1.47·37-s + 0.480·39-s + 1.40·41-s − 0.457·43-s − 0.447·45-s + 1.02·47-s + 1/7·49-s − 0.560·51-s + 0.549·53-s − 1.61·55-s − 0.781·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177744\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(1419.29\)
Root analytic conductor: \(37.6735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{177744} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177744,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.35273079963413, −12.71144173660426, −12.22633632739393, −11.97889765574520, −11.77845859825899, −11.08644190313723, −10.72418471647092, −10.09053374488713, −9.602781522461193, −9.243059383377205, −8.598093773198056, −7.991518405449900, −7.528870344140348, −7.315149439920165, −6.479797848510180, −6.267979516526776, −5.644771647463380, −4.856970774833689, −4.494087872898235, −3.988148250081257, −3.520995569627304, −2.873311654223309, −2.262540909247421, −1.119794769329386, −0.8572165019335201, 0, 0.8572165019335201, 1.119794769329386, 2.262540909247421, 2.873311654223309, 3.520995569627304, 3.988148250081257, 4.494087872898235, 4.856970774833689, 5.644771647463380, 6.267979516526776, 6.479797848510180, 7.315149439920165, 7.528870344140348, 7.991518405449900, 8.598093773198056, 9.243059383377205, 9.602781522461193, 10.09053374488713, 10.72418471647092, 11.08644190313723, 11.77845859825899, 11.97889765574520, 12.22633632739393, 12.71144173660426, 13.35273079963413

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