Properties

Degree 2
Conductor $ 2^{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  − 2·3-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  =  0
Selberg data  =  $(2,\ 128,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9713362299$
$L(\frac12)$  $\approx$  $0.9713362299$
$L(\frac{3}{2})$   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

−18.43686115199919, −17.91683221521600, −17.19630853481417, −16.62475050766921, −15.21399296017509, −14.26636928037331, −13.38742730478456, −12.03375355269583, −11.28853583105346, −10.57265847830630, −9.215939702261382, −7.984207807149970, −6.398714986185465, −5.591008331383187, −4.433244049720489, −1.737036854852305, 1.737036854852305, 4.433244049720489, 5.591008331383187, 6.398714986185465, 7.984207807149970, 9.215939702261382, 10.57265847830630, 11.28853583105346, 12.03375355269583, 13.38742730478456, 14.26636928037331, 15.21399296017509, 16.62475050766921, 17.19630853481417, 17.91683221521600, 18.43686115199919

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