Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.640 - 0.767i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−980. − 294. i)2-s − 8.63e4·3-s + (8.74e5 + 5.77e5i)4-s + 1.68e7i·5-s + (8.47e7 + 2.54e7i)6-s − 3.64e8i·7-s + (−6.87e8 − 8.24e8i)8-s + 3.97e9·9-s + (4.95e9 − 1.64e10i)10-s + 2.95e10·11-s + (−7.55e10 − 4.99e10i)12-s + 1.06e11i·13-s + (−1.07e11 + 3.57e11i)14-s − 1.45e12i·15-s + (4.31e11 + 1.01e12i)16-s + 2.31e12·17-s + ⋯
L(s)  = 1  + (−0.957 − 0.287i)2-s − 1.46·3-s + (0.834 + 0.551i)4-s + 1.72i·5-s + (1.40 + 0.420i)6-s − 1.29i·7-s + (−0.640 − 0.767i)8-s + 1.14·9-s + (0.495 − 1.64i)10-s + 1.13·11-s + (−1.22 − 0.806i)12-s + 0.772i·13-s + (−0.371 + 1.23i)14-s − 2.51i·15-s + (0.392 + 0.919i)16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.640 - 0.767i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ -0.640 - 0.767i)$
$L(\frac{21}{2})$  $\approx$  $0.188245 + 0.402205i$
$L(\frac12)$  $\approx$  $0.188245 + 0.402205i$
$L(11)$   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 \( 1 + (980. + 294. i)T \)
good3 \( 1 + 8.63e4T + 3.48e9T^{2} \)
5 \( 1 - 1.68e7iT - 9.53e13T^{2} \)
7 \( 1 + 3.64e8iT - 7.97e16T^{2} \)
11 \( 1 - 2.95e10T + 6.72e20T^{2} \)
13 \( 1 - 1.06e11iT - 1.90e22T^{2} \)
17 \( 1 - 2.31e12T + 4.06e24T^{2} \)
19 \( 1 + 8.42e11T + 3.75e25T^{2} \)
23 \( 1 - 2.92e12iT - 1.71e27T^{2} \)
29 \( 1 - 2.89e14iT - 1.76e29T^{2} \)
31 \( 1 - 6.94e14iT - 6.71e29T^{2} \)
37 \( 1 + 2.39e15iT - 2.31e31T^{2} \)
41 \( 1 + 8.44e15T + 1.80e32T^{2} \)
43 \( 1 + 2.60e16T + 4.67e32T^{2} \)
47 \( 1 - 4.58e16iT - 2.76e33T^{2} \)
53 \( 1 + 2.23e17iT - 3.05e34T^{2} \)
59 \( 1 + 3.72e17T + 2.61e35T^{2} \)
61 \( 1 - 1.14e18iT - 5.08e35T^{2} \)
67 \( 1 - 1.69e18T + 3.32e36T^{2} \)
71 \( 1 + 1.43e18iT - 1.05e37T^{2} \)
73 \( 1 - 6.55e17T + 1.84e37T^{2} \)
79 \( 1 - 3.74e18iT - 8.96e37T^{2} \)
83 \( 1 + 1.51e19T + 2.40e38T^{2} \)
89 \( 1 - 2.12e19T + 9.72e38T^{2} \)
97 \( 1 + 6.94e19T + 5.43e39T^{2} \)
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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

−17.31024497958422928764758959475, −16.45405265344272193562898594134, −14.39523418008083126052420726396, −11.84917707531714722635520816699, −10.92399935423118521031756166401, −10.06602410386693715534806562257, −7.15187777628180084017804297398, −6.48605787053374240220359758911, −3.59614688516186630005315904042, −1.25987562334497532641399650103, 0.32023787401404535968313928193, 1.39973290110716683551363838174, 5.18907315140848402539942339524, 6.03893162010361670827607475198, 8.373718672991066697883438778754, 9.668584071939141020311182123331, 11.70279572809230743307085832942, 12.35138307509973918794495725401, 15.43163588099115974785097950452, 16.69372627704221466980895497994

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