Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.719 + 0.694i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.62 − 5.69i)2-s + 8.49·3-s + (−0.753 − 63.9i)4-s + 59.7i·5-s + (47.7 − 48.3i)6-s + 483. i·7-s + (−368. − 355. i)8-s − 656.·9-s + (339. + 335. i)10-s + 1.41e3·11-s + (−6.39 − 543. i)12-s − 3.45e3i·13-s + (2.75e3 + 2.71e3i)14-s + 507. i·15-s + (−4.09e3 + 96.3i)16-s − 3.05e3·17-s + ⋯
L(s)  = 1  + (0.702 − 0.711i)2-s + 0.314·3-s + (−0.0117 − 0.999i)4-s + 0.477i·5-s + (0.221 − 0.223i)6-s + 1.40i·7-s + (−0.719 − 0.694i)8-s − 0.901·9-s + (0.339 + 0.335i)10-s + 1.06·11-s + (−0.00370 − 0.314i)12-s − 1.57i·13-s + (1.00 + 0.991i)14-s + 0.150i·15-s + (−0.999 + 0.0235i)16-s − 0.622·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.719 + 0.694i$
motivic weight  =  \(6\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :3),\ 0.719 + 0.694i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.58950 - 0.642019i\)
\(L(\frac12)\)  \(\approx\)  \(1.58950 - 0.642019i\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) 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 \( 1 + (-5.62 + 5.69i)T \)
good3 \( 1 - 8.49T + 729T^{2} \)
5 \( 1 - 59.7iT - 1.56e4T^{2} \)
7 \( 1 - 483. iT - 1.17e5T^{2} \)
11 \( 1 - 1.41e3T + 1.77e6T^{2} \)
13 \( 1 + 3.45e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.05e3T + 2.41e7T^{2} \)
19 \( 1 - 968.T + 4.70e7T^{2} \)
23 \( 1 - 3.31e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.63e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.71e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.60e4iT - 2.56e9T^{2} \)
41 \( 1 + 6.86e3T + 4.75e9T^{2} \)
43 \( 1 - 9.28e4T + 6.32e9T^{2} \)
47 \( 1 - 1.59e5iT - 1.07e10T^{2} \)
53 \( 1 - 8.66e4iT - 2.21e10T^{2} \)
59 \( 1 - 1.28e5T + 4.21e10T^{2} \)
61 \( 1 + 1.89e5iT - 5.15e10T^{2} \)
67 \( 1 + 3.19e5T + 9.04e10T^{2} \)
71 \( 1 + 1.96e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.39e4T + 1.51e11T^{2} \)
79 \( 1 - 1.64e5iT - 2.43e11T^{2} \)
83 \( 1 + 8.02e5T + 3.26e11T^{2} \)
89 \( 1 + 5.41e4T + 4.96e11T^{2} \)
97 \( 1 + 1.10e6T + 8.32e11T^{2} \)
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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

−20.43895082774187658945759071496, −19.33887413499209159501562941774, −17.94420474076352694947796568551, −15.33488768624734424137414583545, −14.39316917820394788122429598033, −12.55555373117196914310731018643, −11.14144158935495434278459082971, −9.055539380151302611188845939990, −5.80367130656092429931878280249, −2.86446596448982736560703191197, 4.20051198457975790597031324160, 6.81522779244751174684062919760, 8.813205526723579910742409612834, 11.64637614916148065835557843715, 13.60154628238059635476301142235, 14.44910702253910699477665917260, 16.50805040251573655748792634686, 17.22426564408722689944198315511, 19.73963874020841646067506370023, 20.84424410033755037557886917399

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