Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 131 $
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-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s + 11-s − 12-s − 2·13-s − 3·14-s + 15-s + 16-s − 5·17-s + 18-s − 4·19-s − 20-s + 3·21-s + 22-s + 3·23-s − 24-s − 4·25-s − 2·26-s − 27-s − 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.654·21-s + 0.213·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-s − 0.392·26-s − 0.192·27-s − 0.566·28-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  =  1
Selberg data  =  $(2,\ 786,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−19.78032103536135, −19.07867276194411, −18.35733294683997, −17.21246791007890, −16.76064635228907, −16.03927464730271, −15.21930514705817, −14.85194382088016, −13.55345097655239, −12.96969028939298, −12.47996714430771, −11.43026347604589, −11.05703042490452, −9.906569290565014, −9.221339067401329, −7.946987484560757, −6.878583331118149, −6.423394789822661, −5.389094787776614, −4.324481394453949, −3.538098425660079, −2.182998701723784, 0, 2.182998701723784, 3.538098425660079, 4.324481394453949, 5.389094787776614, 6.423394789822661, 6.878583331118149, 7.946987484560757, 9.221339067401329, 9.906569290565014, 11.05703042490452, 11.43026347604589, 12.47996714430771, 12.96969028939298, 13.55345097655239, 14.85194382088016, 15.21930514705817, 16.03927464730271, 16.76064635228907, 17.21246791007890, 18.35733294683997, 19.07867276194411, 19.78032103536135

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