Properties

Degree 2
Conductor $ 2^{6} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 3·9-s − 6·13-s + 2·17-s − 25-s + 10·29-s + 2·37-s + 10·41-s − 6·45-s − 7·49-s − 14·53-s + 10·61-s − 12·65-s − 6·73-s + 9·81-s + 4·85-s + 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 18·117-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s − 1.66·13-s + 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.328·37-s + 1.56·41-s − 0.894·45-s − 49-s − 1.92·53-s + 1.28·61-s − 1.48·65-s − 0.702·73-s + 81-s + 0.433·85-s + 1.05·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.66·117-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(64\)    =    \(2^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{64} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 64,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9270373386$
$L(\frac12)$  $\approx$  $0.9270373386$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−19.93016928972141, −19.17924008725702, −17.63226515593639, −17.33116792549253, −16.11526573364450, −14.59250078389593, −14.09304813859310, −12.72487277908245, −11.66649219374340, −10.23421229877844, −9.317356556158666, −7.865740559985357, −6.272820806468766, −5.014522975961726, −2.638951045531797, 2.638951045531797, 5.014522975961726, 6.272820806468766, 7.865740559985357, 9.317356556158666, 10.23421229877844, 11.66649219374340, 12.72487277908245, 14.09304813859310, 14.59250078389593, 16.11526573364450, 17.33116792549253, 17.63226515593639, 19.17924008725702, 19.93016928972141

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