Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.946 - 0.321i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (214. + 291. i)2-s − 1.63e3i·3-s + (−3.93e4 + 1.25e5i)4-s − 1.21e6i·5-s + (4.78e5 − 3.50e5i)6-s + 1.76e7·7-s + (−4.49e7 + 1.52e7i)8-s + 1.26e8·9-s + (3.53e8 − 2.59e8i)10-s − 7.26e8i·11-s + (2.04e8 + 6.45e7i)12-s + 3.16e9i·13-s + (3.78e9 + 5.15e9i)14-s − 1.98e9·15-s + (−1.40e10 − 9.84e9i)16-s + 4.56e10·17-s + ⋯
L(s)  = 1  + (0.591 + 0.806i)2-s − 0.144i·3-s + (−0.300 + 0.953i)4-s − 1.38i·5-s + (0.116 − 0.0852i)6-s + 1.15·7-s + (−0.946 + 0.321i)8-s + 0.979·9-s + (1.11 − 0.821i)10-s − 1.02i·11-s + (0.137 + 0.0433i)12-s + 1.07i·13-s + (0.684 + 0.933i)14-s − 0.200·15-s + (−0.819 − 0.573i)16-s + 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.946 - 0.321i$
motivic weight  =  \(17\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :17/2),\ 0.946 - 0.321i)\)
\(L(9)\)  \(\approx\)  \(2.80285 + 0.463457i\)
\(L(\frac12)\)  \(\approx\)  \(2.80285 + 0.463457i\)
\(L(\frac{19}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) 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 \( 1 + (-214. - 291. i)T \)
good3 \( 1 + 1.63e3iT - 1.29e8T^{2} \)
5 \( 1 + 1.21e6iT - 7.62e11T^{2} \)
7 \( 1 - 1.76e7T + 2.32e14T^{2} \)
11 \( 1 + 7.26e8iT - 5.05e17T^{2} \)
13 \( 1 - 3.16e9iT - 8.65e18T^{2} \)
17 \( 1 - 4.56e10T + 8.27e20T^{2} \)
19 \( 1 - 1.55e10iT - 5.48e21T^{2} \)
23 \( 1 - 4.09e11T + 1.41e23T^{2} \)
29 \( 1 + 3.11e12iT - 7.25e24T^{2} \)
31 \( 1 + 6.57e12T + 2.25e25T^{2} \)
37 \( 1 + 1.36e13iT - 4.56e26T^{2} \)
41 \( 1 + 2.81e12T + 2.61e27T^{2} \)
43 \( 1 - 1.36e13iT - 5.87e27T^{2} \)
47 \( 1 - 6.84e11T + 2.66e28T^{2} \)
53 \( 1 - 2.47e14iT - 2.05e29T^{2} \)
59 \( 1 - 2.03e15iT - 1.27e30T^{2} \)
61 \( 1 + 8.92e14iT - 2.24e30T^{2} \)
67 \( 1 - 5.06e15iT - 1.10e31T^{2} \)
71 \( 1 + 6.85e15T + 2.96e31T^{2} \)
73 \( 1 - 1.37e15T + 4.74e31T^{2} \)
79 \( 1 + 1.78e16T + 1.81e32T^{2} \)
83 \( 1 + 1.48e15iT - 4.21e32T^{2} \)
89 \( 1 + 3.78e16T + 1.37e33T^{2} \)
97 \( 1 - 1.28e17T + 5.95e33T^{2} \)
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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

−16.99965950796119684063262926226, −16.19046681719199722145904300627, −14.44946467525105132783946737515, −13.10569203900022895362774629896, −11.80828628978965990945765676568, −8.937628343226631797641346645253, −7.64516827751068327876161195192, −5.46108413888062433498790504866, −4.21246567400417795451611041452, −1.22236482801359392616534505069, 1.55303549721333137406206445400, 3.26197578022000518802863522119, 5.06729710721355264964753170359, 7.27188846027929701636748326058, 10.04526349373719668620134977526, 10.99749764604405298783714336737, 12.63776667942222238467590938079, 14.51045013819637865408031993025, 15.11654949181115693012658063703, 17.93864695660536674749189565311

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