Properties

Label 2-92400-1.1-c1-0-102
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 − 3·13-s − 5·17-s + 19-s − 21-s − 27-s − 6·29-s + 33-s + 5·37-s + 3·39-s + 11·41-s − 8·43-s + 8·47-s + 49-s + 5·51-s + 53-s − 57-s − 10·59-s − 61-s + 63-s − 11·67-s + 9·71-s − 73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s − 1.21·17-s + 0.229·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 0.174·33-s + 0.821·37-s + 0.480·39-s + 1.71·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.700·51-s + 0.137·53-s − 0.132·57-s − 1.30·59-s − 0.128·61-s + 0.125·63-s − 1.34·67-s + 1.06·71-s − 0.117·73-s − 0.113·77-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: $\chi_{92400} (1, \cdot )$
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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.04479854822877, −13.55688769320381, −12.99962144949660, −12.66083909893354, −12.07721876480071, −11.57381230568859, −11.14190079447250, −10.71990556639809, −10.24951378925584, −9.503783382776373, −9.262150240267846, −8.623710970691560, −7.916934782183702, −7.469073234066693, −7.098407432490733, −6.363339404529136, −5.895440931843818, −5.355120858363459, −4.695376871835566, −4.409775604142152, −3.706641595266392, −2.852583334173181, −2.272045132478418, −1.684642940407048, −0.7510209423956649, 0, 0.7510209423956649, 1.684642940407048, 2.272045132478418, 2.852583334173181, 3.706641595266392, 4.409775604142152, 4.695376871835566, 5.355120858363459, 5.895440931843818, 6.363339404529136, 7.098407432490733, 7.469073234066693, 7.916934782183702, 8.623710970691560, 9.262150240267846, 9.503783382776373, 10.24951378925584, 10.71990556639809, 11.14190079447250, 11.57381230568859, 12.07721876480071, 12.66083909893354, 12.99962144949660, 13.55688769320381, 14.04479854822877

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