Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.212 - 0.977i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (43.9 − 10.8i)2-s + (221. + 221. i)3-s + (1.81e3 − 949. i)4-s + (−7.91e3 + 7.91e3i)5-s + (1.21e4 + 7.34e3i)6-s + 7.68e4i·7-s + (6.94e4 − 6.13e4i)8-s − 7.90e4i·9-s + (−2.62e5 + 4.33e5i)10-s + (−1.97e5 + 1.97e5i)11-s + (6.12e5 + 1.91e5i)12-s + (−4.68e4 − 4.68e4i)13-s + (8.30e5 + 3.37e6i)14-s − 3.50e6·15-s + (2.39e6 − 3.44e6i)16-s + 9.74e6·17-s + ⋯
L(s)  = 1  + (0.971 − 0.238i)2-s + (0.526 + 0.526i)3-s + (0.885 − 0.463i)4-s + (−1.13 + 1.13i)5-s + (0.636 + 0.385i)6-s + 1.72i·7-s + (0.749 − 0.661i)8-s − 0.446i·9-s + (−0.829 + 1.37i)10-s + (−0.370 + 0.370i)11-s + (0.710 + 0.222i)12-s + (−0.0350 − 0.0350i)13-s + (0.412 + 1.67i)14-s − 1.19·15-s + (0.569 − 0.821i)16-s + 1.66·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.212 - 0.977i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.212 - 0.977i)$
$L(6)$  $\approx$  $2.41824 + 1.94957i$
$L(\frac12)$  $\approx$  $2.41824 + 1.94957i$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-43.9 + 10.8i)T \)
good3 \( 1 + (-221. - 221. i)T + 1.77e5iT^{2} \)
5 \( 1 + (7.91e3 - 7.91e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 7.68e4iT - 1.97e9T^{2} \)
11 \( 1 + (1.97e5 - 1.97e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (4.68e4 + 4.68e4i)T + 1.79e12iT^{2} \)
17 \( 1 - 9.74e6T + 3.42e13T^{2} \)
19 \( 1 + (-2.85e6 - 2.85e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 3.12e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.26e8 + 1.26e8i)T + 1.22e16iT^{2} \)
31 \( 1 - 3.46e7T + 2.54e16T^{2} \)
37 \( 1 + (-2.25e8 + 2.25e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 4.55e8iT - 5.50e17T^{2} \)
43 \( 1 + (-7.03e8 + 7.03e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 6.32e8T + 2.47e18T^{2} \)
53 \( 1 + (1.37e9 - 1.37e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-1.10e9 + 1.10e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-3.14e9 - 3.14e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (3.21e9 + 3.21e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.72e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.53e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.44e10T + 7.47e20T^{2} \)
83 \( 1 + (1.61e10 + 1.61e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 1.60e10iT - 2.77e21T^{2} \)
97 \( 1 - 7.16e9T + 7.15e21T^{2} \)
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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

−15.77726208226583979046094568844, −15.17388281584597589981332956691, −14.49218609331961073825115118109, −12.32262286972320345003627527845, −11.50552136287595374658119953173, −9.776722065859260632553988079858, −7.67203944317481421353285994794, −5.76060532789772543570671514564, −3.68988816558517949406720314529, −2.67701747207352616616659873139, 1.01294305109356082414147350480, 3.53242023440833999753727614531, 4.86266919652525414408886891838, 7.38674837004413450008298241633, 8.055682782640770383846653859027, 10.85265750250515289107193592122, 12.43887890093278237991412860234, 13.38440429291529395300576988148, 14.45994954693646001424328319890, 16.31850496691452351538282201153

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