Properties

Degree 2
Conductor $ 2^{4} \cdot 11 $
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 + 5-s + 2·7-s − 2·9-s − 11-s + 4·13-s + 15-s − 2·17-s + 2·21-s + 23-s − 4·25-s − 5·27-s − 7·31-s − 33-s + 2·35-s + 3·37-s + 4·39-s − 8·41-s + 6·43-s − 2·45-s − 8·47-s − 3·49-s − 2·51-s − 6·53-s − 55-s − 5·59-s + 12·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.208·23-s − 4/5·25-s − 0.962·27-s − 1.25·31-s − 0.174·33-s + 0.338·35-s + 0.493·37-s + 0.640·39-s − 1.24·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.134·55-s − 0.650·59-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(176\)    =    \(2^{4} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{176} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 176,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.458816616$
$L(\frac12)$  $\approx$  $1.458816616$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−18.96234428060461, −18.03249379674997, −17.42810920768149, −16.34331292708779, −15.32611240664076, −14.46437983368131, −13.73198911397801, −12.95575045127536, −11.50606278511501, −10.88023677523056, −9.531323384279074, −8.595888941067366, −7.818938261632754, −6.293236729241947, −5.167960041901215, −3.576328582284719, −2.011687025646050, 2.011687025646050, 3.576328582284719, 5.167960041901215, 6.293236729241947, 7.818938261632754, 8.595888941067366, 9.531323384279074, 10.88023677523056, 11.50606278511501, 12.95575045127536, 13.73198911397801, 14.46437983368131, 15.32611240664076, 16.34331292708779, 17.42810920768149, 18.03249379674997, 18.96234428060461

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