Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.114 - 0.993i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−39.1 + 1.02e3i)2-s − 6.15e4·3-s + (−1.04e6 − 8.00e4i)4-s − 2.85e6i·5-s + (2.40e6 − 6.29e7i)6-s − 1.54e8i·7-s + (1.22e8 − 1.06e9i)8-s + 3.02e8·9-s + (2.92e9 + 1.11e8i)10-s + 1.20e10·11-s + (6.43e10 + 4.92e9i)12-s + 1.32e10i·13-s + (1.58e11 + 6.05e9i)14-s + 1.75e11i·15-s + (1.08e12 + 1.67e11i)16-s − 1.51e12·17-s + ⋯
L(s)  = 1  + (−0.0382 + 0.999i)2-s − 1.04·3-s + (−0.997 − 0.0763i)4-s − 0.292i·5-s + (0.0398 − 1.04i)6-s − 0.547i·7-s + (0.114 − 0.993i)8-s + 0.0868·9-s + (0.292 + 0.0111i)10-s + 0.465·11-s + (1.03 + 0.0796i)12-s + 0.0961i·13-s + (0.547 + 0.0209i)14-s + 0.305i·15-s + (0.988 + 0.152i)16-s − 0.752·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.114 - 0.993i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ 0.114 - 0.993i)$
$L(\frac{21}{2})$  $\approx$  $0.638504 + 0.569188i$
$L(\frac12)$  $\approx$  $0.638504 + 0.569188i$
$L(11)$   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 \( 1 + (39.1 - 1.02e3i)T \)
good3 \( 1 + 6.15e4T + 3.48e9T^{2} \)
5 \( 1 + 2.85e6iT - 9.53e13T^{2} \)
7 \( 1 + 1.54e8iT - 7.97e16T^{2} \)
11 \( 1 - 1.20e10T + 6.72e20T^{2} \)
13 \( 1 - 1.32e10iT - 1.90e22T^{2} \)
17 \( 1 + 1.51e12T + 4.06e24T^{2} \)
19 \( 1 + 5.95e12T + 3.75e25T^{2} \)
23 \( 1 - 5.68e13iT - 1.71e27T^{2} \)
29 \( 1 + 9.55e13iT - 1.76e29T^{2} \)
31 \( 1 - 1.17e15iT - 6.71e29T^{2} \)
37 \( 1 + 8.09e15iT - 2.31e31T^{2} \)
41 \( 1 - 1.78e16T + 1.80e32T^{2} \)
43 \( 1 - 2.76e16T + 4.67e32T^{2} \)
47 \( 1 + 1.55e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.45e17iT - 3.05e34T^{2} \)
59 \( 1 - 4.21e17T + 2.61e35T^{2} \)
61 \( 1 + 5.24e17iT - 5.08e35T^{2} \)
67 \( 1 + 2.46e17T + 3.32e36T^{2} \)
71 \( 1 - 3.30e18iT - 1.05e37T^{2} \)
73 \( 1 + 3.00e18T + 1.84e37T^{2} \)
79 \( 1 - 1.13e19iT - 8.96e37T^{2} \)
83 \( 1 + 4.84e18T + 2.40e38T^{2} \)
89 \( 1 - 1.43e19T + 9.72e38T^{2} \)
97 \( 1 - 2.08e18T + 5.43e39T^{2} \)
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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

−17.04364269400868273423305556579, −15.90975038975236702776460227599, −14.21416932086920539594218445264, −12.63668830702007075911798144482, −10.83286399104344739309753096758, −8.943805932464655852674813241842, −7.04900349219617389750887897285, −5.72490247754426836590423496212, −4.28802188612065284077129417342, −0.803367619662063441917762232673, 0.56480692835002547254947116101, 2.48616065117148326012596020780, 4.53584990304748603445082281753, 6.18136812910242702705142384780, 8.771524835211925689591798996619, 10.56798803784212168739874858456, 11.59658290205360089368221557948, 12.80059788991009010129645009826, 14.67587090340326542426544558283, 16.80039217786022184738245743026

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