Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7 $
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 + 4-s + 2·5-s − 7-s − 8-s − 2·10-s + 4·11-s + 6·13-s + 14-s + 16-s − 2·17-s − 4·19-s + 2·20-s − 4·22-s − 8·23-s − 25-s − 6·26-s − 28-s + 2·29-s − 32-s + 2·34-s − 2·35-s − 10·37-s + 4·38-s − 2·40-s + 6·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.371·29-s − 0.176·32-s + 0.342·34-s − 0.338·35-s − 1.64·37-s + 0.648·38-s − 0.316·40-s + 0.937·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{126} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 126,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9104737368$
$L(\frac12)$  $\approx$  $0.9104737368$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.77117040259924, −18.93373372750873, −17.92159314124527, −17.37789158429939, −16.34711419608210, −15.57707994022367, −14.23038534683502, −13.48577504303345, −12.25720048996309, −11.12015185557949, −10.13805865622749, −9.164662289745379, −8.337652051623380, −6.588137121575448, −6.018996477438010, −3.838631711101342, −1.793653735408206, 1.793653735408206, 3.838631711101342, 6.018996477438010, 6.588137121575448, 8.337652051623380, 9.164662289745379, 10.13805865622749, 11.12015185557949, 12.25720048996309, 13.48577504303345, 14.23038534683502, 15.57707994022367, 16.34711419608210, 17.37789158429939, 17.92159314124527, 18.93373372750873, 19.77117040259924

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