Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.982 - 0.187i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−200. + 301. i)2-s − 1.34e4i·3-s + (−5.07e4 − 1.20e5i)4-s − 1.59e6i·5-s + (4.06e6 + 2.70e6i)6-s − 1.66e7·7-s + (4.66e7 + 8.91e6i)8-s − 5.26e7·9-s + (4.80e8 + 3.19e8i)10-s + 6.79e8i·11-s + (−1.62e9 + 6.84e8i)12-s − 3.13e9i·13-s + (3.32e9 − 5.00e9i)14-s − 2.14e10·15-s + (−1.20e10 + 1.22e10i)16-s − 1.26e10·17-s + ⋯
L(s)  = 1  + (−0.553 + 0.832i)2-s − 1.18i·3-s + (−0.387 − 0.921i)4-s − 1.82i·5-s + (0.988 + 0.656i)6-s − 1.08·7-s + (0.982 + 0.187i)8-s − 0.407·9-s + (1.51 + 1.00i)10-s + 0.955i·11-s + (−1.09 + 0.459i)12-s − 1.06i·13-s + (0.602 − 0.906i)14-s − 2.16·15-s + (−0.700 + 0.714i)16-s − 0.440·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.982 - 0.187i$
motivic weight  =  \(17\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :17/2),\ -0.982 - 0.187i)\)
\(L(9)\)  \(\approx\)  \(0.0533047 + 0.562565i\)
\(L(\frac12)\)  \(\approx\)  \(0.0533047 + 0.562565i\)
\(L(\frac{19}{2})\)   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 \( 1 + (200. - 301. i)T \)
good3 \( 1 + 1.34e4iT - 1.29e8T^{2} \)
5 \( 1 + 1.59e6iT - 7.62e11T^{2} \)
7 \( 1 + 1.66e7T + 2.32e14T^{2} \)
11 \( 1 - 6.79e8iT - 5.05e17T^{2} \)
13 \( 1 + 3.13e9iT - 8.65e18T^{2} \)
17 \( 1 + 1.26e10T + 8.27e20T^{2} \)
19 \( 1 - 5.87e9iT - 5.48e21T^{2} \)
23 \( 1 - 5.43e11T + 1.41e23T^{2} \)
29 \( 1 - 2.82e12iT - 7.25e24T^{2} \)
31 \( 1 - 8.42e11T + 2.25e25T^{2} \)
37 \( 1 - 6.17e12iT - 4.56e26T^{2} \)
41 \( 1 + 6.38e13T + 2.61e27T^{2} \)
43 \( 1 - 5.41e13iT - 5.87e27T^{2} \)
47 \( 1 + 3.89e13T + 2.66e28T^{2} \)
53 \( 1 + 5.92e14iT - 2.05e29T^{2} \)
59 \( 1 + 7.11e14iT - 1.27e30T^{2} \)
61 \( 1 + 9.01e14iT - 2.24e30T^{2} \)
67 \( 1 - 8.96e14iT - 1.10e31T^{2} \)
71 \( 1 - 8.49e13T + 2.96e31T^{2} \)
73 \( 1 - 1.01e16T + 4.74e31T^{2} \)
79 \( 1 - 3.99e15T + 1.81e32T^{2} \)
83 \( 1 + 2.19e15iT - 4.21e32T^{2} \)
89 \( 1 + 6.74e16T + 1.37e33T^{2} \)
97 \( 1 + 1.67e15T + 5.95e33T^{2} \)
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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

−16.80730121902065887771577463969, −15.54662487880410782314190218653, −13.17623118777747015540708229049, −12.66791841868302015744680785973, −9.625946911014670383882510287767, −8.264009619480325492515380801668, −6.79829910045218252234204226791, −5.08286517710857280005061792811, −1.40750216629179864349843802112, −0.29246881389725511467881259021, 2.80710324285546941622272471045, 3.79970428476851254744366383718, 6.79851578780200193348641887051, 9.284415707041162343667719877504, 10.40471563913043203567805861550, 11.32477330192434723387097351598, 13.70087527187094083329120225517, 15.41086428626393880439518429185, 16.75720607538679795631736769814, 18.64790310223132575316490930469

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