Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.850 + 0.525i$
Motivic weight 20
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−340. − 965. i)2-s + 8.43e4·3-s + (−8.16e5 + 6.57e5i)4-s + 4.86e6i·5-s + (−2.87e7 − 8.14e7i)6-s − 1.26e8i·7-s + (9.13e8 + 5.64e8i)8-s + 3.62e9·9-s + (4.69e9 − 1.65e9i)10-s + 1.51e10·11-s + (−6.88e10 + 5.54e10i)12-s + 1.95e11i·13-s + (−1.21e11 + 4.29e10i)14-s + 4.10e11i·15-s + (2.34e11 − 1.07e12i)16-s + 3.56e12·17-s + ⋯
L(s)  = 1  + (−0.332 − 0.943i)2-s + 1.42·3-s + (−0.778 + 0.627i)4-s + 0.498i·5-s + (−0.474 − 1.34i)6-s − 0.446i·7-s + (0.850 + 0.525i)8-s + 1.03·9-s + (0.469 − 0.165i)10-s + 0.583·11-s + (−1.11 + 0.895i)12-s + 1.41i·13-s + (−0.421 + 0.148i)14-s + 0.711i·15-s + (0.213 − 0.977i)16-s + 1.76·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.850 + 0.525i$
motivic weight  =  \(20\)
character  :  $\chi_{8} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :10),\ 0.850 + 0.525i)$
$L(\frac{21}{2})$  $\approx$  $2.51272 - 0.714116i$
$L(\frac12)$  $\approx$  $2.51272 - 0.714116i$
$L(11)$   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 \( 1 + (340. + 965. i)T \)
good3 \( 1 - 8.43e4T + 3.48e9T^{2} \)
5 \( 1 - 4.86e6iT - 9.53e13T^{2} \)
7 \( 1 + 1.26e8iT - 7.97e16T^{2} \)
11 \( 1 - 1.51e10T + 6.72e20T^{2} \)
13 \( 1 - 1.95e11iT - 1.90e22T^{2} \)
17 \( 1 - 3.56e12T + 4.06e24T^{2} \)
19 \( 1 - 3.48e12T + 3.75e25T^{2} \)
23 \( 1 + 1.17e13iT - 1.71e27T^{2} \)
29 \( 1 + 7.56e14iT - 1.76e29T^{2} \)
31 \( 1 - 1.11e15iT - 6.71e29T^{2} \)
37 \( 1 + 7.17e14iT - 2.31e31T^{2} \)
41 \( 1 - 3.23e15T + 1.80e32T^{2} \)
43 \( 1 + 1.36e16T + 4.67e32T^{2} \)
47 \( 1 + 4.28e16iT - 2.76e33T^{2} \)
53 \( 1 - 3.34e17iT - 3.05e34T^{2} \)
59 \( 1 - 2.74e17T + 2.61e35T^{2} \)
61 \( 1 - 1.49e17iT - 5.08e35T^{2} \)
67 \( 1 + 2.92e18T + 3.32e36T^{2} \)
71 \( 1 - 3.51e18iT - 1.05e37T^{2} \)
73 \( 1 - 5.45e17T + 1.84e37T^{2} \)
79 \( 1 + 1.64e19iT - 8.96e37T^{2} \)
83 \( 1 + 1.30e19T + 2.40e38T^{2} \)
89 \( 1 + 2.43e19T + 9.72e38T^{2} \)
97 \( 1 + 7.85e18T + 5.43e39T^{2} \)
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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

−16.75681419618032316504737852139, −14.45558277694796179686062361880, −13.77688976317723705720346445304, −11.90294728653814031493149450679, −10.07537199165357462307468122967, −8.897340787422030731625755083306, −7.43551553591885492693753410403, −4.00444262517932716200079227902, −2.85758093021244954972255414480, −1.37926410338343589501915301311, 1.13687142724146467263033837589, 3.33327699434728423491382952218, 5.42079266171210107754440508280, 7.62448288409276365339211490817, 8.648411960657775375516226670179, 9.811196337167676032973743646658, 12.82394180006584504188961957166, 14.28005952985831181980020003994, 15.15781128693331871096030808209, 16.60034003563437831675210637521

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