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Properties

Label 2-92400-1.1-c1-0-124
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 + 4·17-s − 3·19-s + 21-s + 2·23-s − 27-s + 5·29-s − 33-s − 9·37-s + 39-s + 10·43-s + 5·47-s + 49-s − 4·51-s + 6·53-s + 3·57-s − 13·59-s + 6·61-s − 63-s − 67-s − 2·69-s + 9·73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.970·17-s − 0.688·19-s + 0.218·21-s + 0.417·23-s − 0.192·27-s + 0.928·29-s − 0.174·33-s − 1.47·37-s + 0.160·39-s + 1.52·43-s + 0.729·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s + 0.397·57-s − 1.69·59-s + 0.768·61-s − 0.125·63-s − 0.122·67-s − 0.240·69-s + 1.05·73-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: 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 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.11199963759909, −13.67832361106094, −12.89073161873015, −12.55832289768325, −12.16736547232463, −11.75157931604661, −11.03656256559851, −10.67007285829552, −10.12184035754821, −9.785256574615297, −9.025866237253270, −8.730937244841902, −7.994297358154360, −7.444919578635769, −6.905274752416535, −6.516778216696146, −5.780564547024633, −5.507809570527018, −4.770748916313734, −4.232348582756556, −3.658261628855638, −2.962185947905340, −2.367048187899000, −1.481920956846385, −0.8609280521336465, 0, 0.8609280521336465, 1.481920956846385, 2.367048187899000, 2.962185947905340, 3.658261628855638, 4.232348582756556, 4.770748916313734, 5.507809570527018, 5.780564547024633, 6.516778216696146, 6.905274752416535, 7.444919578635769, 7.994297358154360, 8.730937244841902, 9.025866237253270, 9.785256574615297, 10.12184035754821, 10.67007285829552, 11.03656256559851, 11.75157931604661, 12.16736547232463, 12.55832289768325, 12.89073161873015, 13.67832361106094, 14.11199963759909

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