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 + 5-s + 6-s + 3·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s + 4·13-s + 3·14-s + 15-s + 16-s − 7·17-s + 18-s + 20-s + 3·21-s − 3·22-s − 23-s + 24-s − 4·25-s + 4·26-s + 27-s + 3·28-s + 30-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 − 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 0.223·20-s + 0.654·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.566·28-s + 0.182·30-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  =  $(2,\ 786,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.280672363$
$L(\frac12)$  $\approx$  $3.280672363$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
131 \( 1 - T \)
good5 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.56513311202298, −18.36399659796380, −18.06735882124305, −17.24548445997494, −16.16597345256266, −15.43372079278249, −15.03239590921538, −13.95466987383705, −13.57791792371683, −13.02326529834393, −11.90310008019862, −11.04968767647155, −10.56450799702409, −9.373316090770424, −8.414634231802839, −7.845716953444888, −6.707237513733571, −5.770329041826618, −4.787727495881102, −3.974198997826222, −2.606903227563990, −1.712396521719073, 1.712396521719073, 2.606903227563990, 3.974198997826222, 4.787727495881102, 5.770329041826618, 6.707237513733571, 7.845716953444888, 8.414634231802839, 9.373316090770424, 10.56450799702409, 11.04968767647155, 11.90310008019862, 13.02326529834393, 13.57791792371683, 13.95466987383705, 15.03239590921538, 15.43372079278249, 16.16597345256266, 17.24548445997494, 18.06735882124305, 18.36399659796380, 19.56513311202298

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