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 + 7-s + 8-s − 4·13-s + 14-s + 16-s − 6·17-s + 2·19-s − 5·25-s − 4·26-s + 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 2·37-s + 2·38-s − 6·41-s + 8·43-s + 12·47-s + 49-s − 5·50-s − 4·52-s − 6·53-s + 56-s + 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.707·50-s − 0.554·52-s − 0.824·53-s + 0.133·56-s + 0.787·58-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  =  $(2,\ 126,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.530545448$
$L(\frac12)$  $\approx$  $1.530545448$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.98543388915904, −19.22815645736323, −17.89217793495364, −17.22146833902262, −16.00017911981171, −15.23380331079723, −14.27798924734175, −13.45852272915612, −12.37171571172373, −11.52629384186915, −10.47027853370114, −9.207333736191082, −7.803650996170568, −6.713101277728754, −5.319210593790605, −4.191021024052512, −2.391018623478919, 2.391018623478919, 4.191021024052512, 5.319210593790605, 6.713101277728754, 7.803650996170568, 9.207333736191082, 10.47027853370114, 11.52629384186915, 12.37171571172373, 13.45852272915612, 14.27798924734175, 15.23380331079723, 16.00017911981171, 17.22146833902262, 17.89217793495364, 19.22815645736323, 19.98543388915904

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