Properties

Degree 2
Conductor $ 2^{3} $
Sign $1$
Motivic weight 12
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 64·2-s + 658·3-s + 4.09e3·4-s + 4.21e4·6-s + 2.62e5·8-s − 9.84e4·9-s + 1.92e6·11-s + 2.69e6·12-s + 1.67e7·16-s − 4.52e7·17-s − 6.30e6·18-s − 8.79e7·19-s + 1.23e8·22-s + 1.72e8·24-s + 2.44e8·25-s − 4.14e8·27-s + 1.07e9·32-s + 1.26e9·33-s − 2.89e9·34-s − 4.03e8·36-s − 5.62e9·38-s + 8.62e9·41-s − 7.03e9·43-s + 7.87e9·44-s + 1.10e10·48-s + 1.38e10·49-s + 1.56e10·50-s + ⋯
L(s)  = 1  + 2-s + 0.902·3-s + 4-s + 0.902·6-s + 8-s − 0.185·9-s + 1.08·11-s + 0.902·12-s + 16-s − 1.87·17-s − 0.185·18-s − 1.86·19-s + 1.08·22-s + 0.902·24-s + 25-s − 1.06·27-s + 32-s + 0.979·33-s − 1.87·34-s − 0.185·36-s − 1.86·38-s + 1.81·41-s − 1.11·43-s + 1.08·44-s + 0.902·48-s + 49-s + 50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(12\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :6),\ 1)$
$L(\frac{13}{2})$  $\approx$  $3.69393$
$L(\frac12)$  $\approx$  $3.69393$
$L(7)$   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 \( 1 - p^{6} T \)
good3 \( 1 - 658 T + p^{12} T^{2} \)
5 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
7 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
11 \( 1 - 1923122 T + p^{12} T^{2} \)
13 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
17 \( 1 + 45296062 T + p^{12} T^{2} \)
19 \( 1 + 87931438 T + p^{12} T^{2} \)
23 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
29 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
31 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
37 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
41 \( 1 - 8628259682 T + p^{12} T^{2} \)
43 \( 1 + 7030618702 T + p^{12} T^{2} \)
47 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
53 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
59 \( 1 - 8638314482 T + p^{12} T^{2} \)
61 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
67 \( 1 - 175045819538 T + p^{12} T^{2} \)
71 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
73 \( 1 - 49139489378 T + p^{12} T^{2} \)
79 \( ( 1 - p^{6} T )( 1 + p^{6} T ) \)
83 \( 1 + 192940233262 T + p^{12} T^{2} \)
89 \( 1 + 866326445278 T + p^{12} T^{2} \)
97 \( 1 - 1656488134658 T + p^{12} T^{2} \)
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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

−19.43974110645654414015131187006, −17.04109128083643565512552784604, −15.24361422457252761102133978459, −14.26654802933389496596300436012, −12.92475505505345690870885157108, −11.12901993370592185928552434287, −8.726192547436732794180732503388, −6.57501599807266563744776363529, −4.14611536186358318777621754884, −2.30530367347537028886497845649, 2.30530367347537028886497845649, 4.14611536186358318777621754884, 6.57501599807266563744776363529, 8.726192547436732794180732503388, 11.12901993370592185928552434287, 12.92475505505345690870885157108, 14.26654802933389496596300436012, 15.24361422457252761102133978459, 17.04109128083643565512552784604, 19.43974110645654414015131187006

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