Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.957 + 0.287i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.36 + 20.6i)2-s + 150. i·3-s + (−336. − 385. i)4-s − 292. i·5-s + (−3.09e3 − 1.40e3i)6-s − 9.95e3·7-s + (1.10e4 − 3.32e3i)8-s − 2.84e3·9-s + (6.02e3 + 2.73e3i)10-s + 6.58e4i·11-s + (5.78e4 − 5.05e4i)12-s + 4.49e4i·13-s + (9.31e4 − 2.05e5i)14-s + 4.38e4·15-s + (−3.53e4 + 2.59e5i)16-s − 4.69e5·17-s + ⋯
L(s)  = 1  + (−0.413 + 0.910i)2-s + 1.06i·3-s + (−0.657 − 0.753i)4-s − 0.209i·5-s + (−0.974 − 0.442i)6-s − 1.56·7-s + (0.957 − 0.287i)8-s − 0.144·9-s + (0.190 + 0.0865i)10-s + 1.35i·11-s + (0.805 − 0.703i)12-s + 0.436i·13-s + (0.648 − 1.42i)14-s + 0.223·15-s + (−0.134 + 0.990i)16-s − 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.957 + 0.287i$
motivic weight  =  \(9\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :9/2),\ -0.957 + 0.287i)$
$L(5)$  $\approx$  $0.0985981 - 0.672161i$
$L(\frac12)$  $\approx$  $0.0985981 - 0.672161i$
$L(\frac{11}{2})$   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 \( 1 + (9.36 - 20.6i)T \)
good3 \( 1 - 150. iT - 1.96e4T^{2} \)
5 \( 1 + 292. iT - 1.95e6T^{2} \)
7 \( 1 + 9.95e3T + 4.03e7T^{2} \)
11 \( 1 - 6.58e4iT - 2.35e9T^{2} \)
13 \( 1 - 4.49e4iT - 1.06e10T^{2} \)
17 \( 1 + 4.69e5T + 1.18e11T^{2} \)
19 \( 1 - 4.38e5iT - 3.22e11T^{2} \)
23 \( 1 - 1.14e6T + 1.80e12T^{2} \)
29 \( 1 + 5.39e6iT - 1.45e13T^{2} \)
31 \( 1 - 1.85e6T + 2.64e13T^{2} \)
37 \( 1 - 1.45e7iT - 1.29e14T^{2} \)
41 \( 1 - 5.45e6T + 3.27e14T^{2} \)
43 \( 1 + 5.79e6iT - 5.02e14T^{2} \)
47 \( 1 + 1.69e7T + 1.11e15T^{2} \)
53 \( 1 - 4.94e7iT - 3.29e15T^{2} \)
59 \( 1 + 4.70e7iT - 8.66e15T^{2} \)
61 \( 1 + 7.39e7iT - 1.16e16T^{2} \)
67 \( 1 - 2.37e8iT - 2.72e16T^{2} \)
71 \( 1 + 6.33e7T + 4.58e16T^{2} \)
73 \( 1 + 2.73e7T + 5.88e16T^{2} \)
79 \( 1 + 1.20e8T + 1.19e17T^{2} \)
83 \( 1 + 1.31e8iT - 1.86e17T^{2} \)
89 \( 1 - 6.90e8T + 3.50e17T^{2} \)
97 \( 1 - 1.17e8T + 7.60e17T^{2} \)
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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

−20.35654997087681474831710355049, −18.92058135327997373768219223363, −17.10262548445554492417815430900, −15.96687076876154894450233738908, −15.11543906891193679332604392158, −13.05816374540489534758214111123, −10.14137748426947771812555994469, −9.242654175021908540429993295571, −6.71858881119435499044570943558, −4.45881051517567841731270577497, 0.52850205214568098847237753075, 2.96656224618974369709787245977, 6.82589886433999997973353112969, 8.938761873834235855602559250976, 10.88419262385169586855391988775, 12.73574064298309083274962768521, 13.43118495372681160464466741231, 16.25782636158290861170907587868, 17.96764954421477419307056707701, 19.08551141617500367184553189494

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