Properties

Degree 2
Conductor $ 2^{5} \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 + 3·5-s + 4·7-s + 6·9-s − 5·11-s + 4·13-s − 9·15-s − 6·17-s − 7·19-s − 12·21-s − 3·23-s + 4·25-s − 9·27-s + 3·29-s + 4·31-s + 15·33-s + 12·35-s + 8·37-s − 12·39-s − 6·41-s − 8·43-s + 18·45-s + 6·47-s + 9·49-s + 18·51-s + 53-s − 15·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s + 1.51·7-s + 2·9-s − 1.50·11-s + 1.10·13-s − 2.32·15-s − 1.45·17-s − 1.60·19-s − 2.61·21-s − 0.625·23-s + 4/5·25-s − 1.73·27-s + 0.557·29-s + 0.718·31-s + 2.61·33-s + 2.02·35-s + 1.31·37-s − 1.92·39-s − 0.937·41-s − 1.21·43-s + 2.68·45-s + 0.875·47-s + 9/7·49-s + 2.52·51-s + 0.137·53-s − 2.02·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13792\)    =    \(2^{5} \cdot 431\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{13792} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 13792,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;431\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
431 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
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

−16.77938728403716, −15.87994617451942, −15.45410817472850, −14.99902208740289, −13.88013150201051, −13.69498352485712, −12.89850428938662, −12.70867820817584, −11.66552617875833, −11.23911597732312, −10.83475405631389, −10.32945017493787, −9.995176784409672, −8.849647103501781, −8.354879090174665, −7.723493883288323, −6.617356149871961, −6.351902031982537, −5.759898314910281, −5.113911739541168, −4.715302840201092, −4.105312731832032, −2.441740694491275, −1.959536553916767, −1.151807071572739, 0, 1.151807071572739, 1.959536553916767, 2.441740694491275, 4.105312731832032, 4.715302840201092, 5.113911739541168, 5.759898314910281, 6.351902031982537, 6.617356149871961, 7.723493883288323, 8.354879090174665, 8.849647103501781, 9.995176784409672, 10.32945017493787, 10.83475405631389, 11.23911597732312, 11.66552617875833, 12.70867820817584, 12.89850428938662, 13.69498352485712, 13.88013150201051, 14.99902208740289, 15.45410817472850, 15.87994617451942, 16.77938728403716

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