Properties

Label 2-92400-1.1-c1-0-131
Degree $2$
Conductor $92400$
Sign $-1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 + 7-s + 9-s + 11-s + 4·13-s − 17-s − 3·19-s − 21-s − 23-s − 27-s − 3·29-s + 2·31-s − 33-s − 8·37-s − 4·39-s − 2·41-s − 43-s + 2·47-s + 49-s + 51-s − 11·53-s + 3·57-s − 13·59-s + 5·61-s + 63-s − 14·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.242·17-s − 0.688·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 0.557·29-s + 0.359·31-s − 0.174·33-s − 1.31·37-s − 0.640·39-s − 0.312·41-s − 0.152·43-s + 0.291·47-s + 1/7·49-s + 0.140·51-s − 1.51·53-s + 0.397·57-s − 1.69·59-s + 0.640·61-s + 0.125·63-s − 1.71·67-s + 0.120·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{92400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−13.94244009655385, −13.65554477609845, −13.09609823121315, −12.53249415564837, −12.10155714544764, −11.62479859993963, −11.04717484959560, −10.70879709092705, −10.36618275426308, −9.548012215271240, −9.089663890315392, −8.651628793718763, −7.934050778361991, −7.693338779675718, −6.725851957635732, −6.477993005864082, −6.016209201304687, −5.267244267916580, −4.877700778788819, −4.181000731685295, −3.683743554056035, −3.103938364439852, −2.067078108238059, −1.664213559472310, −0.8761685809302384, 0, 0.8761685809302384, 1.664213559472310, 2.067078108238059, 3.103938364439852, 3.683743554056035, 4.181000731685295, 4.877700778788819, 5.267244267916580, 6.016209201304687, 6.477993005864082, 6.725851957635732, 7.693338779675718, 7.934050778361991, 8.651628793718763, 9.089663890315392, 9.548012215271240, 10.36618275426308, 10.70879709092705, 11.04717484959560, 11.62479859993963, 12.10155714544764, 12.53249415564837, 13.09609823121315, 13.65554477609845, 13.94244009655385

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