Properties

Label 2-92400-1.1-c1-0-141
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 + 13-s − 5·17-s + 8·19-s + 21-s − 23-s − 27-s + 3·29-s − 3·31-s − 33-s + 4·37-s − 39-s + 3·41-s + 9·43-s + 2·47-s + 49-s + 5·51-s + 9·53-s − 8·57-s − 5·59-s − 5·61-s − 63-s − 12·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 1.21·17-s + 1.83·19-s + 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.557·29-s − 0.538·31-s − 0.174·33-s + 0.657·37-s − 0.160·39-s + 0.468·41-s + 1.37·43-s + 0.291·47-s + 1/7·49-s + 0.700·51-s + 1.23·53-s − 1.05·57-s − 0.650·59-s − 0.640·61-s − 0.125·63-s − 1.46·67-s + 0.120·69-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 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.91303374271269, −13.56656590397316, −13.20254721921627, −12.47692617798432, −12.09513594299929, −11.69930536001373, −11.02719776344063, −10.81571281067570, −10.13769119620271, −9.580651160125697, −9.144007237881608, −8.776734885442132, −7.920752143380456, −7.425590422089122, −7.032275633694689, −6.336862765666204, −5.959086772908491, −5.422713245349015, −4.779635999070012, −4.197135665091360, −3.714023681548366, −2.908288919704640, −2.412279038683859, −1.439060645968670, −0.8945693447095747, 0, 0.8945693447095747, 1.439060645968670, 2.412279038683859, 2.908288919704640, 3.714023681548366, 4.197135665091360, 4.779635999070012, 5.422713245349015, 5.959086772908491, 6.336862765666204, 7.032275633694689, 7.425590422089122, 7.920752143380456, 8.776734885442132, 9.144007237881608, 9.580651160125697, 10.13769119620271, 10.81571281067570, 11.02719776344063, 11.69930536001373, 12.09513594299929, 12.47692617798432, 13.20254721921627, 13.56656590397316, 13.91303374271269

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