Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.997 + 0.0643i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (492. − 897. i)2-s + 5.32e4·3-s + (−5.62e5 − 8.84e5i)4-s − 1.65e7i·5-s + (2.62e7 − 4.78e7i)6-s + 1.30e8i·7-s + (−1.07e9 + 6.90e7i)8-s − 6.47e8·9-s + (−1.48e10 − 8.16e9i)10-s + 2.41e10·11-s + (−2.99e10 − 4.71e10i)12-s − 6.71e10i·13-s + (1.16e11 + 6.41e10i)14-s − 8.83e11i·15-s + (−4.66e11 + 9.95e11i)16-s − 2.87e12·17-s + ⋯
L(s)  = 1  + (0.481 − 0.876i)2-s + 0.902·3-s + (−0.536 − 0.843i)4-s − 1.69i·5-s + (0.434 − 0.790i)6-s + 0.460i·7-s + (−0.997 + 0.0643i)8-s − 0.185·9-s + (−1.48 − 0.816i)10-s + 0.932·11-s + (−0.484 − 0.761i)12-s − 0.487i·13-s + (0.404 + 0.221i)14-s − 1.53i·15-s + (−0.423 + 0.905i)16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0643i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.997 + 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.997 + 0.0643i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ -0.997 + 0.0643i)$
$L(\frac{21}{2})$  $\approx$  $0.0827514 - 2.57114i$
$L(\frac12)$  $\approx$  $0.0827514 - 2.57114i$
$L(11)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-492. + 897. i)T \)
good3 \( 1 - 5.32e4T + 3.48e9T^{2} \)
5 \( 1 + 1.65e7iT - 9.53e13T^{2} \)
7 \( 1 - 1.30e8iT - 7.97e16T^{2} \)
11 \( 1 - 2.41e10T + 6.72e20T^{2} \)
13 \( 1 + 6.71e10iT - 1.90e22T^{2} \)
17 \( 1 + 2.87e12T + 4.06e24T^{2} \)
19 \( 1 - 7.21e12T + 3.75e25T^{2} \)
23 \( 1 + 2.96e12iT - 1.71e27T^{2} \)
29 \( 1 + 4.38e14iT - 1.76e29T^{2} \)
31 \( 1 - 9.28e14iT - 6.71e29T^{2} \)
37 \( 1 + 7.26e15iT - 2.31e31T^{2} \)
41 \( 1 + 1.47e15T + 1.80e32T^{2} \)
43 \( 1 - 3.77e16T + 4.67e32T^{2} \)
47 \( 1 + 5.25e16iT - 2.76e33T^{2} \)
53 \( 1 + 2.49e17iT - 3.05e34T^{2} \)
59 \( 1 - 2.43e17T + 2.61e35T^{2} \)
61 \( 1 - 5.14e17iT - 5.08e35T^{2} \)
67 \( 1 - 2.15e18T + 3.32e36T^{2} \)
71 \( 1 - 1.26e18iT - 1.05e37T^{2} \)
73 \( 1 - 3.77e18T + 1.84e37T^{2} \)
79 \( 1 - 2.34e18iT - 8.96e37T^{2} \)
83 \( 1 + 1.71e19T + 2.40e38T^{2} \)
89 \( 1 - 6.55e18T + 9.72e38T^{2} \)
97 \( 1 - 1.29e20T + 5.43e39T^{2} \)
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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

−15.66982034086066641766070211011, −14.07823332576092237718745157960, −12.92471638028541442339177776479, −11.70765677745035093240475593892, −9.328985495671441352682584889870, −8.638926624159304893064916285064, −5.44213968276812057561659141071, −3.95626970103672275846626566286, −2.18530884784956870021133266016, −0.70988937495203352185632857709, 2.71115050930887708228136505817, 3.88892074048458811735393619751, 6.42208885791461242068690364092, 7.48844472144312999135689819728, 9.237081627827857598711834704853, 11.36606109108083434302818476597, 13.76429515643184755280456525613, 14.35587626601478279493706107133, 15.44427932862536050570814693965, 17.32470820111269509945167933111

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