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Properties

Label 2-92400-1.1-c1-0-115
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 $0$

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 + 4·19-s + 21-s + 27-s + 6·29-s + 33-s + 10·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s + 6·51-s − 6·53-s + 4·57-s + 4·59-s − 2·61-s + 63-s − 4·67-s + 16·71-s − 2·73-s + 77-s + 81-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 + 0.917·19-s + 0.218·21-s + 0.192·27-s + 1.11·29-s + 0.174·33-s + 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.125·63-s − 0.488·67-s + 1.89·71-s − 0.234·73-s + 0.113·77-s + 1/9·81-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: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.202271330\)
\(L(\frac12)\) \(\approx\) \(5.202271330\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.80602779695841, −13.59637909248009, −12.83129888751489, −12.27750905416438, −12.06723678010744, −11.23082556166929, −11.03892288498144, −10.25880206055104, −9.772815211527703, −9.430943616504904, −8.847881184575841, −8.182198968895868, −7.881380052704605, −7.463297611220402, −6.713288599444173, −6.223852924629062, −5.560295654906467, −5.114732670773128, −4.338195394487217, −3.916617212793397, −3.153280168669769, −2.827500121768353, −1.971523535769558, −1.163625489974577, −0.8263610132535359, 0.8263610132535359, 1.163625489974577, 1.971523535769558, 2.827500121768353, 3.153280168669769, 3.916617212793397, 4.338195394487217, 5.114732670773128, 5.560295654906467, 6.223852924629062, 6.713288599444173, 7.463297611220402, 7.881380052704605, 8.182198968895868, 8.847881184575841, 9.430943616504904, 9.772815211527703, 10.25880206055104, 11.03892288498144, 11.23082556166929, 12.06723678010744, 12.27750905416438, 12.83129888751489, 13.59637909248009, 13.80602779695841

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