Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.409 - 0.912i$
Motivic weight 13
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.6 − 89.6i)2-s + 86.8i·3-s + (−7.87e3 − 2.27e3i)4-s + 4.55e4i·5-s + (7.78e3 + 1.10e3i)6-s − 2.49e5·7-s + (−3.03e5 + 6.76e5i)8-s + 1.58e6·9-s + (4.08e6 + 5.77e5i)10-s + 5.28e6i·11-s + (1.97e5 − 6.83e5i)12-s + 1.17e7i·13-s + (−3.15e6 + 2.23e7i)14-s − 3.95e6·15-s + (5.67e7 + 3.57e7i)16-s − 1.22e8·17-s + ⋯
L(s)  = 1  + (0.140 − 0.990i)2-s + 0.0688i·3-s + (−0.960 − 0.277i)4-s + 1.30i·5-s + (0.0681 + 0.00963i)6-s − 0.800·7-s + (−0.409 + 0.912i)8-s + 0.995·9-s + (1.29 + 0.182i)10-s + 0.900i·11-s + (0.0190 − 0.0661i)12-s + 0.675i·13-s + (−0.112 + 0.792i)14-s − 0.0896·15-s + (0.846 + 0.533i)16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(14-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.409 - 0.912i$
motivic weight  =  \(13\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 8,\ (\ :13/2),\ 0.409 - 0.912i)$
$L(7)$  $\approx$  $0.861594 + 0.557836i$
$L(\frac12)$  $\approx$  $0.861594 + 0.557836i$
$L(\frac{15}{2})$   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 \( 1 + (-12.6 + 89.6i)T \)
good3 \( 1 - 86.8iT - 1.59e6T^{2} \)
5 \( 1 - 4.55e4iT - 1.22e9T^{2} \)
7 \( 1 + 2.49e5T + 9.68e10T^{2} \)
11 \( 1 - 5.28e6iT - 3.45e13T^{2} \)
13 \( 1 - 1.17e7iT - 3.02e14T^{2} \)
17 \( 1 + 1.22e8T + 9.90e15T^{2} \)
19 \( 1 + 5.19e7iT - 4.20e16T^{2} \)
23 \( 1 + 1.22e9T + 5.04e17T^{2} \)
29 \( 1 - 4.04e9iT - 1.02e19T^{2} \)
31 \( 1 - 1.61e9T + 2.44e19T^{2} \)
37 \( 1 - 4.09e9iT - 2.43e20T^{2} \)
41 \( 1 - 4.23e10T + 9.25e20T^{2} \)
43 \( 1 + 7.66e10iT - 1.71e21T^{2} \)
47 \( 1 - 6.09e10T + 5.46e21T^{2} \)
53 \( 1 - 2.24e11iT - 2.60e22T^{2} \)
59 \( 1 + 4.64e11iT - 1.04e23T^{2} \)
61 \( 1 + 4.58e10iT - 1.61e23T^{2} \)
67 \( 1 - 8.92e11iT - 5.48e23T^{2} \)
71 \( 1 - 1.53e12T + 1.16e24T^{2} \)
73 \( 1 - 8.01e11T + 1.67e24T^{2} \)
79 \( 1 - 5.92e10T + 4.66e24T^{2} \)
83 \( 1 - 4.57e12iT - 8.87e24T^{2} \)
89 \( 1 + 5.17e12T + 2.19e25T^{2} \)
97 \( 1 + 3.32e12T + 6.73e25T^{2} \)
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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

−18.85814272459657297609529692312, −17.92152618359532937978523820896, −15.52645908863776102809780346645, −13.98639995207428565954205140386, −12.46384751873774000714790628875, −10.74014405608278100508052425132, −9.606402884081694962719894491195, −6.80888680183317884899368065102, −4.04426730356893955415865051031, −2.23587952828443494914035394069, 0.49486991567735139869259407329, 4.25622886693741870371577012967, 6.08447339184230149496102052983, 8.122457366588545647905886740315, 9.603337428135151282561086935765, 12.61816711450467959279398823669, 13.49791924423284230980364770844, 15.70946310279636852723642767406, 16.39114699050864670206393534780, 17.92124200467724311169362870690

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