Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 131 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s + 5·7-s − 8-s + 9-s + 3·10-s + 3·11-s + 12-s + 2·13-s − 5·14-s − 3·15-s + 16-s − 3·17-s − 18-s − 4·19-s − 3·20-s + 5·21-s − 3·22-s − 3·23-s − 24-s + 4·25-s − 2·26-s + 27-s + 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s + 0.554·13-s − 1.33·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s + 1.09·21-s − 0.639·22-s − 0.625·23-s − 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.944·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(786\)    =    \(2 \cdot 3 \cdot 131\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{786} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 786,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.392005187$
$L(\frac12)$  $\approx$  $1.392005187$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;131\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
131 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−19.78103558852743, −19.48456459190158, −18.67622092649224, −17.79688212918202, −17.45618476843721, −16.36676022586209, −15.63843964001434, −15.01753735161807, −14.44724635747353, −13.68309566334872, −12.26659607286900, −11.79833705063636, −11.02267473404027, −10.52012305463675, −8.912370674872703, −8.669967477699423, −7.828225049197172, −7.332903360805183, −6.092167025510846, −4.413375721823584, −4.087120851703879, −2.412053035869580, −1.151475730701919, 1.151475730701919, 2.412053035869580, 4.087120851703879, 4.413375721823584, 6.092167025510846, 7.332903360805183, 7.828225049197172, 8.669967477699423, 8.912370674872703, 10.52012305463675, 11.02267473404027, 11.79833705063636, 12.26659607286900, 13.68309566334872, 14.44724635747353, 15.01753735161807, 15.63843964001434, 16.36676022586209, 17.45618476843721, 17.79688212918202, 18.67622092649224, 19.48456459190158, 19.78103558852743

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