Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.755 + 0.655i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (351. − 85.4i)2-s − 2.16e4i·3-s + (1.16e5 − 6.01e4i)4-s − 4.65e5i·5-s + (−1.85e6 − 7.62e6i)6-s + 2.24e7·7-s + (3.58e7 − 3.10e7i)8-s − 3.40e8·9-s + (−3.97e7 − 1.63e8i)10-s + 6.06e8i·11-s + (−1.30e9 − 2.52e9i)12-s + 2.12e9i·13-s + (7.90e9 − 1.92e9i)14-s − 1.00e10·15-s + (9.95e9 − 1.40e10i)16-s − 5.45e9·17-s + ⋯
L(s)  = 1  + (0.971 − 0.235i)2-s − 1.90i·3-s + (0.888 − 0.458i)4-s − 0.532i·5-s + (−0.450 − 1.85i)6-s + 1.47·7-s + (0.755 − 0.655i)8-s − 2.64·9-s + (−0.125 − 0.517i)10-s + 0.852i·11-s + (−0.875 − 1.69i)12-s + 0.723i·13-s + (1.43 − 0.347i)14-s − 1.01·15-s + (0.579 − 0.815i)16-s − 0.189·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.755 + 0.655i$
motivic weight  =  \(17\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :17/2),\ -0.755 + 0.655i)\)
\(L(9)\)  \(\approx\)  \(1.26913 - 3.39936i\)
\(L(\frac12)\)  \(\approx\)  \(1.26913 - 3.39936i\)
\(L(\frac{19}{2})\)   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 \( 1 + (-351. + 85.4i)T \)
good3 \( 1 + 2.16e4iT - 1.29e8T^{2} \)
5 \( 1 + 4.65e5iT - 7.62e11T^{2} \)
7 \( 1 - 2.24e7T + 2.32e14T^{2} \)
11 \( 1 - 6.06e8iT - 5.05e17T^{2} \)
13 \( 1 - 2.12e9iT - 8.65e18T^{2} \)
17 \( 1 + 5.45e9T + 8.27e20T^{2} \)
19 \( 1 - 5.72e9iT - 5.48e21T^{2} \)
23 \( 1 + 1.30e11T + 1.41e23T^{2} \)
29 \( 1 + 4.23e11iT - 7.25e24T^{2} \)
31 \( 1 - 3.83e12T + 2.25e25T^{2} \)
37 \( 1 - 2.39e13iT - 4.56e26T^{2} \)
41 \( 1 + 5.79e12T + 2.61e27T^{2} \)
43 \( 1 + 4.00e13iT - 5.87e27T^{2} \)
47 \( 1 + 1.56e14T + 2.66e28T^{2} \)
53 \( 1 + 6.45e14iT - 2.05e29T^{2} \)
59 \( 1 - 8.37e14iT - 1.27e30T^{2} \)
61 \( 1 + 3.98e14iT - 2.24e30T^{2} \)
67 \( 1 + 5.22e15iT - 1.10e31T^{2} \)
71 \( 1 - 5.50e15T + 2.96e31T^{2} \)
73 \( 1 - 1.47e15T + 4.74e31T^{2} \)
79 \( 1 + 1.94e16T + 1.81e32T^{2} \)
83 \( 1 - 3.36e16iT - 4.21e32T^{2} \)
89 \( 1 + 1.09e16T + 1.37e33T^{2} \)
97 \( 1 - 6.57e16T + 5.95e33T^{2} \)
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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

−17.14521164387078950779221875841, −14.66915209597362234092984109013, −13.61152720937922548278526875488, −12.34045131537348145858511006523, −11.46716168609139856792588099002, −8.094945654931231735816200992178, −6.72685988738649562245573095167, −4.94962375485526704063620624150, −2.12566203620866197403036355806, −1.24440228023970873025986676931, 2.97717704296000622773981362798, 4.41603815869647545164432837856, 5.58555376286536589481944204469, 8.321749936454324893083221149588, 10.62520328622677354075236051212, 11.39352233033989317655097531386, 14.17069348483854868562415177262, 14.92980906238282173924764824514, 16.04540243462560870140386960123, 17.41217349147772092662841350557

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