↑

Properties

Label 2-193200-1.1-c1-0-162
Degree $2$
Conductor $193200$
Sign $-1$
Analytic cond. $1542.70$
Root an. cond. $39.2773$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s + 3·11-s + 4·13-s − 8·17-s − 3·19-s + 21-s + 23-s + 27-s + 10·29-s − 4·31-s + 3·33-s + 4·39-s − 7·41-s + 6·43-s − 5·47-s + 49-s − 8·51-s − 13·53-s − 3·57-s − 12·59-s + 10·61-s + 63-s + 14·67-s + 69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s − 1.94·17-s − 0.688·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.522·33-s + 0.640·39-s − 1.09·41-s + 0.914·43-s − 0.729·47-s + 1/7·49-s − 1.12·51-s − 1.78·53-s − 0.397·57-s − 1.56·59-s + 1.28·61-s + 0.125·63-s + 1.71·67-s + 0.120·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1542.70\)
Root analytic conductor: \(39.2773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 193200,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.44027359144499, −12.78212656394882, −12.61039833803381, −11.87009947381743, −11.34434144433678, −11.00163353378086, −10.64254566214771, −10.02490502503379, −9.330474088001926, −9.004093118834025, −8.684295815802440, −8.033994274055353, −7.914991296079856, −6.748918914011505, −6.576984347431926, −6.474971576451427, −5.466299918012643, −4.893097758027558, −4.328199595833681, −3.991480087857453, −3.401564828047591, −2.724068070970815, −2.116166421187639, −1.578015497128920, −0.9779908949807907, 0, 0.9779908949807907, 1.578015497128920, 2.116166421187639, 2.724068070970815, 3.401564828047591, 3.991480087857453, 4.328199595833681, 4.893097758027558, 5.466299918012643, 6.474971576451427, 6.576984347431926, 6.748918914011505, 7.914991296079856, 8.033994274055353, 8.684295815802440, 9.004093118834025, 9.330474088001926, 10.02490502503379, 10.64254566214771, 11.00163353378086, 11.34434144433678, 11.87009947381743, 12.61039833803381, 12.78212656394882, 13.44027359144499

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