Properties

Degree 2
Conductor $ 2^{6} \cdot 5^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·7-s + 9-s + 4·13-s + 4·19-s + 8·21-s − 6·23-s − 4·27-s − 6·29-s − 8·31-s + 2·37-s + 8·39-s − 6·41-s + 8·43-s + 6·47-s + 9·49-s − 6·53-s + 8·57-s − 12·59-s + 2·61-s + 4·63-s + 10·67-s − 12·69-s + 12·71-s − 16·73-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s + 0.917·19-s + 1.74·21-s − 1.25·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + 1.22·67-s − 1.44·69-s + 1.42·71-s − 1.87·73-s + 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{193600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 193600,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.56025356542833, −12.97536777183197, −12.39433602303451, −11.90345230478932, −11.31666416564915, −11.03502113035927, −10.67406872751058, −9.892909567677809, −9.313737015350381, −9.110917474431207, −8.462561819519349, −8.131767556429586, −7.636034497559044, −7.481570001106743, −6.646786227159146, −5.906588220937866, −5.526212318537752, −5.067047327775435, −4.218984976280917, −3.885220357713478, −3.436154215551848, −2.714735556415924, −2.064894744672264, −1.658558737295685, −1.124139622344984, 0, 1.124139622344984, 1.658558737295685, 2.064894744672264, 2.714735556415924, 3.436154215551848, 3.885220357713478, 4.218984976280917, 5.067047327775435, 5.526212318537752, 5.906588220937866, 6.646786227159146, 7.481570001106743, 7.636034497559044, 8.131767556429586, 8.462561819519349, 9.110917474431207, 9.313737015350381, 9.892909567677809, 10.67406872751058, 11.03502113035927, 11.31666416564915, 11.90345230478932, 12.39433602303451, 12.97536777183197, 13.56025356542833

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