Properties

Label 2-92400-1.1-c1-0-195
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 + 2·13-s − 17-s + 7·19-s − 21-s + 5·23-s + 27-s + 3·29-s + 4·31-s − 33-s + 2·37-s + 2·39-s − 12·41-s − 43-s − 4·47-s + 49-s − 51-s + 53-s + 7·57-s − 9·59-s − 11·61-s − 63-s + 2·67-s + 5·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.242·17-s + 1.60·19-s − 0.218·21-s + 1.04·23-s + 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 1.87·41-s − 0.152·43-s − 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.137·53-s + 0.927·57-s − 1.17·59-s − 1.40·61-s − 0.125·63-s + 0.244·67-s + 0.601·69-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.93313514950750, −13.51345649375307, −13.36043211733128, −12.67859904546311, −12.08563572345512, −11.73633903633723, −11.06916016021960, −10.61884935578259, −10.01433788330860, −9.591913660219178, −9.150422264995458, −8.523489775156005, −8.163566867467800, −7.538215351507101, −7.014031728890989, −6.559104902067825, −5.940327519294265, −5.245802148484235, −4.805422577684341, −4.163337868465072, −3.234537759511368, −3.199950593914747, −2.486467834846924, −1.518681850686104, −1.070491344638488, 0, 1.070491344638488, 1.518681850686104, 2.486467834846924, 3.199950593914747, 3.234537759511368, 4.163337868465072, 4.805422577684341, 5.245802148484235, 5.940327519294265, 6.559104902067825, 7.014031728890989, 7.538215351507101, 8.163566867467800, 8.523489775156005, 9.150422264995458, 9.591913660219178, 10.01433788330860, 10.61884935578259, 11.06916016021960, 11.73633903633723, 12.08563572345512, 12.67859904546311, 13.36043211733128, 13.51345649375307, 13.93313514950750

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