Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.151 - 0.988i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.3 − 361. i)2-s − 1.37e4i·3-s + (−1.30e5 − 1.32e4i)4-s − 9.63e4i·5-s + (−4.98e6 − 2.52e5i)6-s − 1.47e7·7-s + (−7.18e6 + 4.69e7i)8-s − 6.09e7·9-s + (−3.48e7 − 1.76e6i)10-s − 6.68e8i·11-s + (−1.82e8 + 1.79e9i)12-s + 1.72e8i·13-s + (−2.70e8 + 5.34e9i)14-s − 1.32e9·15-s + (1.68e10 + 3.45e9i)16-s + 1.92e10·17-s + ⋯
L(s)  = 1  + (0.0506 − 0.998i)2-s − 1.21i·3-s + (−0.994 − 0.101i)4-s − 0.110i·5-s + (−1.21 − 0.0614i)6-s − 0.968·7-s + (−0.151 + 0.988i)8-s − 0.471·9-s + (−0.110 − 0.00558i)10-s − 0.940i·11-s + (−0.122 + 1.20i)12-s + 0.0586i·13-s + (−0.0490 + 0.967i)14-s − 0.133·15-s + (0.979 + 0.201i)16-s + 0.670·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.151 - 0.988i$
motivic weight  =  \(17\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :17/2),\ 0.151 - 0.988i)\)
\(L(9)\)  \(\approx\)  \(0.380443 + 0.326608i\)
\(L(\frac12)\)  \(\approx\)  \(0.380443 + 0.326608i\)
\(L(\frac{19}{2})\)   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 \( 1 + (-18.3 + 361. i)T \)
good3 \( 1 + 1.37e4iT - 1.29e8T^{2} \)
5 \( 1 + 9.63e4iT - 7.62e11T^{2} \)
7 \( 1 + 1.47e7T + 2.32e14T^{2} \)
11 \( 1 + 6.68e8iT - 5.05e17T^{2} \)
13 \( 1 - 1.72e8iT - 8.65e18T^{2} \)
17 \( 1 - 1.92e10T + 8.27e20T^{2} \)
19 \( 1 - 1.11e11iT - 5.48e21T^{2} \)
23 \( 1 + 5.92e11T + 1.41e23T^{2} \)
29 \( 1 + 2.95e12iT - 7.25e24T^{2} \)
31 \( 1 + 7.53e12T + 2.25e25T^{2} \)
37 \( 1 - 2.20e13iT - 4.56e26T^{2} \)
41 \( 1 + 8.98e13T + 2.61e27T^{2} \)
43 \( 1 - 1.45e13iT - 5.87e27T^{2} \)
47 \( 1 + 7.43e13T + 2.66e28T^{2} \)
53 \( 1 - 2.84e14iT - 2.05e29T^{2} \)
59 \( 1 + 1.09e15iT - 1.27e30T^{2} \)
61 \( 1 - 4.14e14iT - 2.24e30T^{2} \)
67 \( 1 + 2.20e15iT - 1.10e31T^{2} \)
71 \( 1 - 6.67e15T + 2.96e31T^{2} \)
73 \( 1 + 1.64e15T + 4.74e31T^{2} \)
79 \( 1 + 1.46e16T + 1.81e32T^{2} \)
83 \( 1 + 3.05e16iT - 4.21e32T^{2} \)
89 \( 1 - 4.48e16T + 1.37e33T^{2} \)
97 \( 1 + 3.26e16T + 5.95e33T^{2} \)
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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

−16.62832298438798941207006927397, −14.05324530748978345124170353268, −12.92744958645437688459521354627, −11.93696406677571895003340451513, −10.01036312218474303064552878407, −8.139165633037932135793941659408, −6.04199474848134139651486125987, −3.42236100638088223131837215769, −1.65474362341665800312596460035, −0.20320152641273845558803230875, 3.62996626297516112666301053419, 5.10400842994735521196470916937, 6.98011079676199296523866415224, 9.153808162504719291976464075546, 10.19506816984228163348265593649, 12.79966982979552979630045640538, 14.61109220538226011968104997709, 15.72691412683607312843554773335, 16.58898140348359010503737009919, 18.18335585410968004628893464992

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