Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 3·5-s − 2·6-s − 3·7-s + 9-s + 6·10-s − 5·11-s + 2·12-s + 2·13-s + 6·14-s − 3·15-s − 4·16-s − 6·17-s − 2·18-s − 6·19-s − 6·20-s − 3·21-s + 10·22-s + 9·23-s + 4·25-s − 4·26-s + 27-s − 6·28-s + 29-s + 6·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s − 1.13·7-s + 1/3·9-s + 1.89·10-s − 1.50·11-s + 0.577·12-s + 0.554·13-s + 1.60·14-s − 0.774·15-s − 16-s − 1.45·17-s − 0.471·18-s − 1.37·19-s − 1.34·20-s − 0.654·21-s + 2.13·22-s + 1.87·23-s + 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.13·28-s + 0.185·29-s + 1.09·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(141\)    =    \(3 \cdot 47\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{141} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 141,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
47 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.69559980596455, −19.02174980945884, −18.63032201093019, −17.46161326554361, −16.30849502709337, −15.71591156392934, −15.16807302019726, −13.29822223725169, −12.78597999000325, −11.06455012201967, −10.59810682607065, −9.222291061463395, −8.537254750564664, −7.632494910065733, −6.710082450442274, −4.364669351247937, −2.779086768331929, 0, 2.779086768331929, 4.364669351247937, 6.710082450442274, 7.632494910065733, 8.537254750564664, 9.222291061463395, 10.59810682607065, 11.06455012201967, 12.78597999000325, 13.29822223725169, 15.16807302019726, 15.71591156392934, 16.30849502709337, 17.46161326554361, 18.63032201093019, 19.02174980945884, 19.69559980596455

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