Properties

Label 2-51600-1.1-c1-0-23
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
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 + 3·7-s + 9-s − 5·13-s + 4·17-s + 7·19-s − 3·21-s − 4·23-s − 27-s + 3·29-s − 3·31-s + 4·37-s + 5·39-s − 11·41-s + 43-s − 2·47-s + 2·49-s − 4·51-s − 10·53-s − 7·57-s + 4·59-s − 11·61-s + 3·63-s + 67-s + 4·69-s + 14·71-s + 3·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s − 1.38·13-s + 0.970·17-s + 1.60·19-s − 0.654·21-s − 0.834·23-s − 0.192·27-s + 0.557·29-s − 0.538·31-s + 0.657·37-s + 0.800·39-s − 1.71·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s − 0.560·51-s − 1.37·53-s − 0.927·57-s + 0.520·59-s − 1.40·61-s + 0.377·63-s + 0.122·67-s + 0.481·69-s + 1.66·71-s + 0.351·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.877515684\)
\(L(\frac12)\) \(\approx\) \(1.877515684\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 - T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.40906037003660, −14.08008114401069, −13.62780459087264, −12.76597420263369, −12.23088252308026, −12.02583382674738, −11.33439137141287, −11.15699921161496, −10.19796208897208, −9.910351250186290, −9.526478638760298, −8.680041677867386, −7.937167788362951, −7.744644384855845, −7.157975081771095, −6.553960630254434, −5.691848283931110, −5.340311482324270, −4.823426334878694, −4.368266148602139, −3.428388654594990, −2.872378243589648, −1.905138496087442, −1.414948772393753, −0.5105473533061555, 0.5105473533061555, 1.414948772393753, 1.905138496087442, 2.872378243589648, 3.428388654594990, 4.368266148602139, 4.823426334878694, 5.340311482324270, 5.691848283931110, 6.553960630254434, 7.157975081771095, 7.744644384855845, 7.937167788362951, 8.680041677867386, 9.526478638760298, 9.910351250186290, 10.19796208897208, 11.15699921161496, 11.33439137141287, 12.02583382674738, 12.23088252308026, 12.76597420263369, 13.62780459087264, 14.08008114401069, 14.40906037003660

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