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Properties

Label 2-92400-1.1-c1-0-134
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 − 6·13-s − 2·17-s − 4·19-s + 21-s − 4·23-s + 27-s − 2·29-s − 4·31-s + 33-s + 2·37-s − 6·39-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 2·51-s + 2·53-s − 4·57-s + 8·59-s + 10·61-s + 63-s − 8·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s + 1.04·59-s + 1.28·61-s + 0.125·63-s − 0.977·67-s − 0.481·69-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: Trivial
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
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

−14.10438523632466, −13.75373046326887, −13.04215610526611, −12.57718975128213, −12.26242971645206, −11.68340797360920, −11.06139823222214, −10.64194844255269, −10.06059331525148, −9.490543443070669, −9.205564341565925, −8.588808940866837, −7.952572392547955, −7.695982771342027, −6.910755566969163, −6.744144410300972, −5.752432629020765, −5.406985997071465, −4.609366246776605, −4.158070139753355, −3.763484764252920, −2.749619474383137, −2.287223913835693, −1.930426589485604, −0.8804129791939598, 0, 0.8804129791939598, 1.930426589485604, 2.287223913835693, 2.749619474383137, 3.763484764252920, 4.158070139753355, 4.609366246776605, 5.406985997071465, 5.752432629020765, 6.744144410300972, 6.910755566969163, 7.695982771342027, 7.952572392547955, 8.588808940866837, 9.205564341565925, 9.490543443070669, 10.06059331525148, 10.64194844255269, 11.06139823222214, 11.68340797360920, 12.26242971645206, 12.57718975128213, 13.04215610526611, 13.75373046326887, 14.10438523632466

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