Properties

Label 2-92400-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s + 11-s − 4·13-s − 6·17-s − 4·19-s − 21-s − 6·23-s − 27-s + 6·31-s − 33-s − 8·37-s + 4·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s + 6·51-s + 6·53-s + 4·57-s + 4·59-s + 6·61-s + 63-s − 12·67-s + 6·69-s + 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.07·31-s − 0.174·33-s − 1.31·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s + 0.768·61-s + 0.125·63-s − 1.46·67-s + 0.722·69-s + 0.468·73-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: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2243561112\)
\(L(\frac12)\) \(\approx\) \(0.2243561112\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−13.78671610014046, −13.24954919669264, −12.94931321202615, −12.14471166908734, −11.81324212985709, −11.62828854493975, −10.79380128403914, −10.40286571797769, −10.01436550825680, −9.438729911130860, −8.705366823927839, −8.425272633052654, −7.800719222407641, −7.107480633238130, −6.676602442277688, −6.303319204437209, −5.608275151269552, −4.896085552784563, −4.647124307800486, −4.034218781168687, −3.388003471904620, −2.358660441035922, −2.092459287773304, −1.308483902863889, −0.1534892566645327, 0.1534892566645327, 1.308483902863889, 2.092459287773304, 2.358660441035922, 3.388003471904620, 4.034218781168687, 4.647124307800486, 4.896085552784563, 5.608275151269552, 6.303319204437209, 6.676602442277688, 7.107480633238130, 7.800719222407641, 8.425272633052654, 8.705366823927839, 9.438729911130860, 10.01436550825680, 10.40286571797769, 10.79380128403914, 11.62828854493975, 11.81324212985709, 12.14471166908734, 12.94931321202615, 13.24954919669264, 13.78671610014046

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