Properties

Degree 2
Conductor $ 47 \cdot 569 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 5·7-s + 3·8-s + 9-s + 2·10-s − 11-s + 2·12-s − 4·13-s + 5·14-s + 4·15-s − 16-s − 3·17-s − 18-s − 5·19-s + 2·20-s + 10·21-s + 22-s − 3·23-s − 6·24-s − 25-s + 4·26-s + 4·27-s + 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.88·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s + 1.33·14-s + 1.03·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.14·19-s + 0.447·20-s + 2.18·21-s + 0.213·22-s − 0.625·23-s − 1.22·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s + 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26743 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(26743\)    =    \(47 \cdot 569\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{26743} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 26743,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{47,\;569\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{47,\;569\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad47 \( 1 + T \)
569 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−16.19749625463088, −15.85654962258102, −15.14143471881911, −14.70392145858624, −13.71990123031011, −13.29271839244716, −12.62719526714702, −12.43538739632992, −11.82570406199275, −11.04352352173562, −10.69504464094628, −10.06166800634527, −9.676412020054801, −9.032835061361824, −8.542735821015412, −7.757742741647074, −7.149128089039270, −6.763154246286237, −6.034191041530998, −5.448585386097418, −4.760645213309887, −4.062023725474041, −3.581846771758994, −2.655036109386631, −1.660589697446375, 0, 0, 0, 1.660589697446375, 2.655036109386631, 3.581846771758994, 4.062023725474041, 4.760645213309887, 5.448585386097418, 6.034191041530998, 6.763154246286237, 7.149128089039270, 7.757742741647074, 8.542735821015412, 9.032835061361824, 9.676412020054801, 10.06166800634527, 10.69504464094628, 11.04352352173562, 11.82570406199275, 12.43538739632992, 12.62719526714702, 13.29271839244716, 13.71990123031011, 14.70392145858624, 15.14143471881911, 15.85654962258102, 16.19749625463088

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