Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{2} \cdot 431 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·7-s + 6·9-s + 5·11-s − 4·13-s + 6·17-s + 7·19-s − 12·21-s − 3·23-s − 9·27-s + 3·29-s − 4·31-s − 15·33-s − 8·37-s + 12·39-s − 6·41-s − 8·43-s + 6·47-s + 9·49-s − 18·51-s − 53-s − 21·57-s + 9·59-s + 2·61-s + 24·63-s + 10·67-s + 9·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.51·7-s + 2·9-s + 1.50·11-s − 1.10·13-s + 1.45·17-s + 1.60·19-s − 2.61·21-s − 0.625·23-s − 1.73·27-s + 0.557·29-s − 0.718·31-s − 2.61·33-s − 1.31·37-s + 1.92·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s − 2.52·51-s − 0.137·53-s − 2.78·57-s + 1.17·59-s + 0.256·61-s + 3.02·63-s + 1.22·67-s + 1.08·69-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 344800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 344800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(344800\)    =    \(2^{5} \cdot 5^{2} \cdot 431\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{344800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 344800,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;431\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
431 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.42292870504539, −12.08717269162366, −11.85004779667089, −11.51862088939653, −11.28475264173243, −10.52407255511992, −10.07286460636460, −9.875271320084783, −9.332776428537338, −8.512108891470928, −8.266392137916639, −7.390734476715495, −7.214834627444709, −6.912637736368010, −6.108802578606174, −5.620135819208136, −5.353856024395500, −4.829447294106376, −4.604186996125455, −3.731490798233382, −3.502392064170723, −2.451759297492933, −1.582865730882699, −1.391976201209274, −0.8568610236647167, 0, 0.8568610236647167, 1.391976201209274, 1.582865730882699, 2.451759297492933, 3.502392064170723, 3.731490798233382, 4.604186996125455, 4.829447294106376, 5.353856024395500, 5.620135819208136, 6.108802578606174, 6.912637736368010, 7.214834627444709, 7.390734476715495, 8.266392137916639, 8.512108891470928, 9.332776428537338, 9.875271320084783, 10.07286460636460, 10.52407255511992, 11.28475264173243, 11.51862088939653, 11.85004779667089, 12.08717269162366, 12.42292870504539

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