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Properties

Label 2-92400-1.1-c1-0-126
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 − 6·17-s − 21-s + 27-s − 2·29-s + 33-s + 2·37-s − 2·39-s − 6·41-s − 4·43-s + 8·47-s + 49-s − 6·51-s + 6·53-s + 10·61-s − 63-s + 8·67-s − 16·71-s + 10·73-s − 77-s − 8·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.218·21-s + 0.192·27-s − 0.371·29-s + 0.174·33-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 1.28·61-s − 0.125·63-s + 0.977·67-s − 1.89·71-s + 1.17·73-s − 0.113·77-s − 0.900·79-s + 1/9·81-s + 1.31·83-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.03207510477350, −13.57587022798030, −13.10736697251031, −12.81384670903657, −12.06021345403914, −11.72401238713566, −11.13153341353942, −10.53113355258649, −10.12175763572010, −9.457955303653753, −9.195398564211206, −8.539994930251041, −8.238765985201386, −7.441339717518335, −6.961042688605471, −6.659004552443692, −5.925653385254443, −5.324671188846983, −4.656644994205697, −4.139313540853744, −3.618432670535508, −2.918727079416475, −2.302330584886371, −1.854736864434101, −0.8661508758121926, 0, 0.8661508758121926, 1.854736864434101, 2.302330584886371, 2.918727079416475, 3.618432670535508, 4.139313540853744, 4.656644994205697, 5.324671188846983, 5.925653385254443, 6.659004552443692, 6.961042688605471, 7.441339717518335, 8.238765985201386, 8.539994930251041, 9.195398564211206, 9.457955303653753, 10.12175763572010, 10.53113355258649, 11.13153341353942, 11.72401238713566, 12.06021345403914, 12.81384670903657, 13.10736697251031, 13.57587022798030, 14.03207510477350

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