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Properties

Label 2-92400-1.1-c1-0-121
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 + 2·17-s − 21-s − 6·23-s − 27-s − 4·29-s − 2·31-s − 33-s − 8·37-s + 6·41-s + 4·43-s + 49-s − 2·51-s + 6·53-s − 12·59-s − 2·61-s + 63-s + 4·67-s + 6·69-s + 16·73-s + 77-s − 10·79-s + 81-s − 14·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.485·17-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 0.742·29-s − 0.359·31-s − 0.174·33-s − 1.31·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.722·69-s + 1.87·73-s + 0.113·77-s − 1.12·79-s + 1/9·81-s − 1.53·83-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.09435636309139, −13.74773185928354, −12.90637312333269, −12.60395824179001, −12.05124211690666, −11.70152433574642, −11.09781736123681, −10.72137944282034, −10.17867883910441, −9.648464639419790, −9.185945260024353, −8.545274869742483, −8.036298580088941, −7.423381274735779, −7.102767275349179, −6.331529338827202, −5.840868363578039, −5.463072155827760, −4.800714995356539, −4.178831242179487, −3.739040285235192, −3.027771751061590, −2.123087135662005, −1.660983726223573, −0.8465471547773424, 0, 0.8465471547773424, 1.660983726223573, 2.123087135662005, 3.027771751061590, 3.739040285235192, 4.178831242179487, 4.800714995356539, 5.463072155827760, 5.840868363578039, 6.331529338827202, 7.102767275349179, 7.423381274735779, 8.036298580088941, 8.545274869742483, 9.185945260024353, 9.648464639419790, 10.17867883910441, 10.72137944282034, 11.09781736123681, 11.70152433574642, 12.05124211690666, 12.60395824179001, 12.90637312333269, 13.74773185928354, 14.09435636309139

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