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Properties

Label 2-92400-1.1-c1-0-171
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 − 8·19-s + 21-s − 6·23-s + 27-s + 6·29-s − 2·31-s + 33-s − 2·37-s − 2·39-s + 8·43-s − 12·47-s + 49-s + 6·51-s − 6·53-s − 8·57-s − 6·59-s + 8·61-s + 63-s + 2·67-s − 6·69-s + 10·73-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 − 1.83·19-s + 0.218·21-s − 1.25·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.320·39-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 1.05·57-s − 0.781·59-s + 1.02·61-s + 0.125·63-s + 0.244·67-s − 0.722·69-s + 1.17·73-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 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.22761616789087, −13.80294888921120, −12.93190186460658, −12.69695838865477, −12.15367717264968, −11.76940138582460, −11.08554019896735, −10.47673776642447, −10.16624743208401, −9.607679052022957, −9.116051823640162, −8.472968627043635, −7.979791004016982, −7.822274513164684, −6.999055763171621, −6.445225479438825, −6.017266886528679, −5.250094948062887, −4.712606183692432, −4.140141792968627, −3.621689662443040, −2.965570883241704, −2.233113692329187, −1.797827740665210, −0.9816304451173875, 0, 0.9816304451173875, 1.797827740665210, 2.233113692329187, 2.965570883241704, 3.621689662443040, 4.140141792968627, 4.712606183692432, 5.250094948062887, 6.017266886528679, 6.445225479438825, 6.999055763171621, 7.822274513164684, 7.979791004016982, 8.472968627043635, 9.116051823640162, 9.607679052022957, 10.16624743208401, 10.47673776642447, 11.08554019896735, 11.76940138582460, 12.15367717264968, 12.69695838865477, 12.93190186460658, 13.80294888921120, 14.22761616789087

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