Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.946 - 0.323i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−31.4 + 32.5i)2-s + (219. − 219. i)3-s + (−64.7 − 2.04e3i)4-s + (−4.57e3 − 4.57e3i)5-s + (221. + 1.40e4i)6-s + 2.77e3i·7-s + (6.85e4 + 6.23e4i)8-s + 8.09e4i·9-s + (2.92e5 − 4.62e3i)10-s + (1.73e4 + 1.73e4i)11-s + (−4.63e5 − 4.34e5i)12-s + (−1.16e6 + 1.16e6i)13-s + (−9.02e4 − 8.74e4i)14-s − 2.00e6·15-s + (−4.18e6 + 2.65e5i)16-s − 7.63e6·17-s + ⋯
L(s)  = 1  + (−0.695 + 0.718i)2-s + (0.521 − 0.521i)3-s + (−0.0316 − 0.999i)4-s + (−0.654 − 0.654i)5-s + (0.0116 + 0.736i)6-s + 0.0624i·7-s + (0.739 + 0.672i)8-s + 0.456i·9-s + (0.925 − 0.0146i)10-s + (0.0324 + 0.0324i)11-s + (−0.537 − 0.504i)12-s + (−0.871 + 0.871i)13-s + (−0.0448 − 0.0434i)14-s − 0.682·15-s + (−0.998 + 0.0631i)16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.946 - 0.323i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.946 - 0.323i)$
$L(6)$  $\approx$  $0.0465976 + 0.280306i$
$L(\frac12)$  $\approx$  $0.0465976 + 0.280306i$
$L(\frac{13}{2})$   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 \( 1 + (31.4 - 32.5i)T \)
good3 \( 1 + (-219. + 219. i)T - 1.77e5iT^{2} \)
5 \( 1 + (4.57e3 + 4.57e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 2.77e3iT - 1.97e9T^{2} \)
11 \( 1 + (-1.73e4 - 1.73e4i)T + 2.85e11iT^{2} \)
13 \( 1 + (1.16e6 - 1.16e6i)T - 1.79e12iT^{2} \)
17 \( 1 + 7.63e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.62e6 + 1.62e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 3.41e7iT - 9.52e14T^{2} \)
29 \( 1 + (5.22e7 - 5.22e7i)T - 1.22e16iT^{2} \)
31 \( 1 + 1.70e8T + 2.54e16T^{2} \)
37 \( 1 + (-1.36e8 - 1.36e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 9.43e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.75e8 + 1.75e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 2.33e9T + 2.47e18T^{2} \)
53 \( 1 + (2.13e9 + 2.13e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (2.73e8 + 2.73e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (-7.10e9 + 7.10e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-6.25e9 + 6.25e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 3.05e9iT - 2.31e20T^{2} \)
73 \( 1 + 3.14e10iT - 3.13e20T^{2} \)
79 \( 1 + 4.76e10T + 7.47e20T^{2} \)
83 \( 1 + (4.66e10 - 4.66e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 1.17e10iT - 2.77e21T^{2} \)
97 \( 1 - 3.59e10T + 7.15e21T^{2} \)
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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

−16.89129593974032714749042196307, −15.86009144698599562008986843159, −14.50780505049124955957988302537, −13.16477836835440327236578235931, −11.34637072883661459107796045496, −9.357048938008969833879812206935, −8.167095316334171049672989656461, −7.00903758893142444840852918991, −4.81930990734606412681323476327, −1.82800152269200506951571052150, 0.13994910323790668179551086072, 2.67818318629679318960022481324, 4.00694129891866700580428329156, 7.21842202348919532624457332449, 8.718472188706325633438695289158, 10.08000265637749388069638107553, 11.30856329681904076687871976439, 12.75684912968735280032103433801, 14.67991901821559545496927679765, 15.77327808550854995755803168298

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