Properties

Degree 2
Conductor $ 2^{7} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 4·7-s + 9-s + 2·11-s − 2·13-s + 4·15-s − 2·17-s − 2·19-s + 8·21-s + 4·23-s − 25-s + 4·27-s + 6·29-s − 4·33-s + 8·35-s − 10·37-s + 4·39-s − 6·41-s − 6·43-s − 2·45-s − 8·47-s + 9·49-s + 4·51-s + 6·53-s − 4·55-s + 4·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s − 0.458·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.696·33-s + 1.35·35-s − 1.64·37-s + 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{128} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 128,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−19.76668259797521, −19.59884082071417, −18.53510149336765, −17.31156570356867, −16.72482907069496, −15.88804349257463, −15.12284818247548, −13.64300623916529, −12.44338350662649, −11.96377224112661, −10.90944990278511, −9.886348706897055, −8.635939205168057, −6.974641220039240, −6.330502600254835, −4.843734315536875, −3.367302696619693, 0, 3.367302696619693, 4.843734315536875, 6.330502600254835, 6.974641220039240, 8.635939205168057, 9.886348706897055, 10.90944990278511, 11.96377224112661, 12.44338350662649, 13.64300623916529, 15.12284818247548, 15.88804349257463, 16.72482907069496, 17.31156570356867, 18.53510149336765, 19.59884082071417, 19.76668259797521

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