Properties

Degree 2
Conductor $ 2^{7} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 2·17-s + 25-s + 2·41-s + 49-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 2·113-s + ⋯
L(s)  = 1  − 9-s − 2·17-s + 25-s + 2·41-s + 49-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 2·113-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{128} (63, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5827916324\)
\(L(\frac12)\)  \(\approx\)  \(0.5827916324\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

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

−13.60653957404905351544532534862, −12.62730928344066147828554111521, −11.40941624187610021847514464408, −10.72859342897654553267618939149, −9.220994759867439580089527677520, −8.461795530674634695923023014770, −7.02072783744560160256463601653, −5.84282968988336850818779137015, −4.41060008842499151557844234715, −2.63592829534740587398498486196, 2.63592829534740587398498486196, 4.41060008842499151557844234715, 5.84282968988336850818779137015, 7.02072783744560160256463601653, 8.461795530674634695923023014770, 9.220994759867439580089527677520, 10.72859342897654553267618939149, 11.40941624187610021847514464408, 12.62730928344066147828554111521, 13.60653957404905351544532534862

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