Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \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  − 3-s + 4·5-s − 7-s + 9-s + 2·11-s − 6·13-s − 4·15-s − 4·17-s − 4·19-s + 21-s + 2·23-s + 11·25-s − 27-s − 2·29-s − 2·33-s − 4·35-s + 2·37-s + 6·39-s − 4·43-s + 4·45-s + 12·47-s + 49-s + 4·51-s − 6·53-s + 8·55-s + 4·57-s − 8·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 1.03·15-s − 0.970·17-s − 0.917·19-s + 0.218·21-s + 0.417·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 0.348·33-s − 0.676·35-s + 0.328·37-s + 0.960·39-s − 0.609·43-s + 0.596·45-s + 1.75·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{84} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 84,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9724619752$
$L(\frac12)$  $\approx$  $0.9724619752$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−19.10038407439374, −17.96916217776553, −17.12059670696826, −16.84909588946188, −15.18560086242639, −14.18271778304861, −13.15717312296324, −12.33374621805755, −10.84044260103234, −9.846442211058849, −9.094314662852253, −6.962073557839571, −6.057004801742162, −4.809169099166338, −2.242397962559298, 2.242397962559298, 4.809169099166338, 6.057004801742162, 6.962073557839571, 9.094314662852253, 9.846442211058849, 10.84044260103234, 12.33374621805755, 13.15717312296324, 14.18271778304861, 15.18560086242639, 16.84909588946188, 17.12059670696826, 17.96916217776553, 19.10038407439374

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