Properties

Degree 2
Conductor $ 2^{3} $
Sign $1$
Motivic weight 6
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 46·3-s + 64·4-s − 368·6-s − 512·8-s + 1.38e3·9-s − 2.33e3·11-s + 2.94e3·12-s + 4.09e3·16-s − 1.72e3·17-s − 1.10e4·18-s − 2.48e3·19-s + 1.87e4·22-s − 2.35e4·24-s + 1.56e4·25-s + 3.02e4·27-s − 3.27e4·32-s − 1.07e5·33-s + 1.38e4·34-s + 8.87e4·36-s + 1.98e4·38-s + 1.34e5·41-s − 7.49e4·43-s − 1.49e5·44-s + 1.88e5·48-s + 1.17e5·49-s − 1.25e5·50-s + ⋯
L(s)  = 1  − 2-s + 1.70·3-s + 4-s − 1.70·6-s − 8-s + 1.90·9-s − 1.75·11-s + 1.70·12-s + 16-s − 0.351·17-s − 1.90·18-s − 0.361·19-s + 1.75·22-s − 1.70·24-s + 25-s + 1.53·27-s − 32-s − 2.99·33-s + 0.351·34-s + 1.90·36-s + 0.361·38-s + 1.95·41-s − 0.942·43-s − 1.75·44-s + 1.70·48-s + 49-s − 50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :3),\ 1)$
$L(\frac{7}{2})$  $\approx$  $1.22549$
$L(\frac12)$  $\approx$  $1.22549$
$L(4)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + p^{3} T \)
good3 \( 1 - 46 T + p^{6} T^{2} \)
5 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
7 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
11 \( 1 + 2338 T + p^{6} T^{2} \)
13 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
17 \( 1 + 1726 T + p^{6} T^{2} \)
19 \( 1 + 2482 T + p^{6} T^{2} \)
23 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
29 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
31 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
37 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
41 \( 1 - 134642 T + p^{6} T^{2} \)
43 \( 1 + 74914 T + p^{6} T^{2} \)
47 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
53 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
59 \( 1 - 304958 T + p^{6} T^{2} \)
61 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
67 \( 1 + 596626 T + p^{6} T^{2} \)
71 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
73 \( 1 + 593134 T + p^{6} T^{2} \)
79 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
83 \( 1 - 678926 T + p^{6} T^{2} \)
89 \( 1 + 357262 T + p^{6} T^{2} \)
97 \( 1 - 1822754 T + p^{6} T^{2} \)
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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

−20.41414734279292018217605937190, −19.21825778876155913017987658786, −18.17122328355539026994403599436, −15.99369080576565762208921123204, −14.85835495689579820061808143356, −13.06078905220351370928828405064, −10.38995241288330938476719449432, −8.813119579607484487899833209497, −7.59772539174242386983133551146, −2.60414127848132862422623311079, 2.60414127848132862422623311079, 7.59772539174242386983133551146, 8.813119579607484487899833209497, 10.38995241288330938476719449432, 13.06078905220351370928828405064, 14.85835495689579820061808143356, 15.99369080576565762208921123204, 18.17122328355539026994403599436, 19.21825778876155913017987658786, 20.41414734279292018217605937190

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