Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.961 + 0.274i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.62 − 6.52i)2-s − 32.4·3-s + (−21.2 + 60.3i)4-s − 199. i·5-s + (150. + 212. i)6-s + 19.6i·7-s + (492. − 140. i)8-s + 326.·9-s + (−1.29e3 + 920. i)10-s − 924.·11-s + (690. − 1.96e3i)12-s − 1.55e3i·13-s + (128. − 90.9i)14-s + 6.46e3i·15-s + (−3.19e3 − 2.56e3i)16-s + 5.14e3·17-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)2-s − 1.20·3-s + (−0.331 + 0.943i)4-s − 1.59i·5-s + (0.695 + 0.982i)6-s + 0.0573i·7-s + (0.961 − 0.274i)8-s + 0.448·9-s + (−1.29 + 0.920i)10-s − 0.694·11-s + (0.399 − 1.13i)12-s − 0.705i·13-s + (0.0467 − 0.0331i)14-s + 1.91i·15-s + (−0.779 − 0.626i)16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.961 + 0.274i$
motivic weight  =  \(6\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :3),\ -0.961 + 0.274i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0620303 - 0.443723i\)
\(L(\frac12)\)  \(\approx\)  \(0.0620303 - 0.443723i\)
\(L(4)\)   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 \( 1 + (4.62 + 6.52i)T \)
good3 \( 1 + 32.4T + 729T^{2} \)
5 \( 1 + 199. iT - 1.56e4T^{2} \)
7 \( 1 - 19.6iT - 1.17e5T^{2} \)
11 \( 1 + 924.T + 1.77e6T^{2} \)
13 \( 1 + 1.55e3iT - 4.82e6T^{2} \)
17 \( 1 - 5.14e3T + 2.41e7T^{2} \)
19 \( 1 + 1.69e3T + 4.70e7T^{2} \)
23 \( 1 + 1.92e4iT - 1.48e8T^{2} \)
29 \( 1 - 1.65e4iT - 5.94e8T^{2} \)
31 \( 1 + 7.55e3iT - 8.87e8T^{2} \)
37 \( 1 - 2.89e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.21e4T + 4.75e9T^{2} \)
43 \( 1 - 5.89e3T + 6.32e9T^{2} \)
47 \( 1 + 6.44e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.97e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.42e5T + 4.21e10T^{2} \)
61 \( 1 - 9.64e4iT - 5.15e10T^{2} \)
67 \( 1 + 7.52e4T + 9.04e10T^{2} \)
71 \( 1 + 5.56e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.85e5T + 1.51e11T^{2} \)
79 \( 1 - 3.42e5iT - 2.43e11T^{2} \)
83 \( 1 + 9.29e5T + 3.26e11T^{2} \)
89 \( 1 - 4.34e5T + 4.96e11T^{2} \)
97 \( 1 - 6.43e5T + 8.32e11T^{2} \)
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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

−20.17239350882098367322195051708, −18.35610090068436920822282142813, −17.04324461773209812419233519878, −16.33304776733959239223497508252, −12.90432594182836474830169553867, −12.04150628738107464748147852586, −10.35341239530777941003521327660, −8.412656806276341346209563536903, −5.08955307076141463251605139114, −0.59900738430399026808425440808, 5.83390581301779880453249431197, 7.28196052882993323425064481501, 10.17149663938200605468979743911, 11.33866024450355041823754287163, 14.12461614241560506729601217816, 15.57600905421142132280584557032, 17.05924221878944068214319727118, 18.16850458107949772546400630618, 19.10045606120582166191202488481, 21.77355723222417296547922522008

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