Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.785 - 0.619i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−998. − 225. i)2-s + 6.01e4·3-s + (9.46e5 + 4.51e5i)4-s + 8.23e6i·5-s + (−6.00e7 − 1.35e7i)6-s + 5.26e8i·7-s + (−8.43e8 − 6.64e8i)8-s + 1.30e8·9-s + (1.86e9 − 8.22e9i)10-s + 6.02e9·11-s + (5.69e10 + 2.71e10i)12-s − 1.38e11i·13-s + (1.19e11 − 5.26e11i)14-s + 4.95e11i·15-s + (6.91e11 + 8.54e11i)16-s − 1.98e12·17-s + ⋯
L(s)  = 1  + (−0.975 − 0.220i)2-s + 1.01·3-s + (0.902 + 0.430i)4-s + 0.843i·5-s + (−0.993 − 0.224i)6-s + 1.86i·7-s + (−0.785 − 0.619i)8-s + 0.0374·9-s + (0.186 − 0.822i)10-s + 0.232·11-s + (0.919 + 0.438i)12-s − 1.00i·13-s + (0.411 − 1.81i)14-s + 0.859i·15-s + (0.629 + 0.777i)16-s − 0.985·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.785 - 0.619i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ -0.785 - 0.619i)$
$L(\frac{21}{2})$  $\approx$  $0.328856 + 0.948349i$
$L(\frac12)$  $\approx$  $0.328856 + 0.948349i$
$L(11)$   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 \( 1 + (998. + 225. i)T \)
good3 \( 1 - 6.01e4T + 3.48e9T^{2} \)
5 \( 1 - 8.23e6iT - 9.53e13T^{2} \)
7 \( 1 - 5.26e8iT - 7.97e16T^{2} \)
11 \( 1 - 6.02e9T + 6.72e20T^{2} \)
13 \( 1 + 1.38e11iT - 1.90e22T^{2} \)
17 \( 1 + 1.98e12T + 4.06e24T^{2} \)
19 \( 1 + 8.88e12T + 3.75e25T^{2} \)
23 \( 1 + 2.01e13iT - 1.71e27T^{2} \)
29 \( 1 + 2.99e14iT - 1.76e29T^{2} \)
31 \( 1 - 1.39e15iT - 6.71e29T^{2} \)
37 \( 1 - 1.32e15iT - 2.31e31T^{2} \)
41 \( 1 - 1.06e16T + 1.80e32T^{2} \)
43 \( 1 + 1.54e15T + 4.67e32T^{2} \)
47 \( 1 - 6.01e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.12e17iT - 3.05e34T^{2} \)
59 \( 1 - 4.77e15T + 2.61e35T^{2} \)
61 \( 1 + 3.65e16iT - 5.08e35T^{2} \)
67 \( 1 - 6.78e17T + 3.32e36T^{2} \)
71 \( 1 + 2.60e18iT - 1.05e37T^{2} \)
73 \( 1 - 2.08e18T + 1.84e37T^{2} \)
79 \( 1 - 4.99e18iT - 8.96e37T^{2} \)
83 \( 1 - 2.79e19T + 2.40e38T^{2} \)
89 \( 1 + 2.07e19T + 9.72e38T^{2} \)
97 \( 1 + 9.97e19T + 5.43e39T^{2} \)
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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

−17.72324180458221932623320452787, −15.51259392137518803926029975830, −14.79663677340744438018749818991, −12.49975596124956687431599185857, −10.86722947953560907215203913682, −9.099498398691068355117886751721, −8.262737722348055310098585587510, −6.30678858167844302006994478587, −2.93532660431001814010410500805, −2.25898271294691408625748531527, 0.40395241721850415079895435193, 1.89266412092521303620052162985, 4.09762691817911038111432140333, 6.88470529323056669711189649782, 8.289737377314714741588855103259, 9.413270478485200462613306584844, 11.03631797238341971720847990376, 13.37356131894560398928003940824, 14.68307787897029078864215212862, 16.50512309014373815276951857543

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