Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.324 - 0.945i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16.6 + 42.0i)2-s + (−491. − 491. i)3-s + (−1.49e3 + 1.40e3i)4-s + (6.77e3 − 6.77e3i)5-s + (1.24e4 − 2.88e4i)6-s + 4.75e4i·7-s + (−8.38e4 − 3.94e4i)8-s + 3.05e5i·9-s + (3.98e5 + 1.72e5i)10-s + (−3.48e5 + 3.48e5i)11-s + (1.42e6 + 4.41e4i)12-s + (6.23e5 + 6.23e5i)13-s + (−1.99e6 + 7.91e5i)14-s − 6.65e6·15-s + (2.60e5 − 4.18e6i)16-s − 4.47e5·17-s + ⋯
L(s)  = 1  + (0.368 + 0.929i)2-s + (−1.16 − 1.16i)3-s + (−0.728 + 0.684i)4-s + (0.969 − 0.969i)5-s + (0.655 − 1.51i)6-s + 1.06i·7-s + (−0.905 − 0.425i)8-s + 1.72i·9-s + (1.25 + 0.544i)10-s + (−0.652 + 0.652i)11-s + (1.64 + 0.0512i)12-s + (0.465 + 0.465i)13-s + (−0.993 + 0.393i)14-s − 2.26·15-s + (0.0620 − 0.998i)16-s − 0.0765·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.324 - 0.945i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.324 - 0.945i)$
$L(6)$  $\approx$  $0.645304 + 0.903699i$
$L(\frac12)$  $\approx$  $0.645304 + 0.903699i$
$L(\frac{13}{2})$   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 \( 1 + (-16.6 - 42.0i)T \)
good3 \( 1 + (491. + 491. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-6.77e3 + 6.77e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 4.75e4iT - 1.97e9T^{2} \)
11 \( 1 + (3.48e5 - 3.48e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-6.23e5 - 6.23e5i)T + 1.79e12iT^{2} \)
17 \( 1 + 4.47e5T + 3.42e13T^{2} \)
19 \( 1 + (-1.39e7 - 1.39e7i)T + 1.16e14iT^{2} \)
23 \( 1 - 4.84e7iT - 9.52e14T^{2} \)
29 \( 1 + (4.44e6 + 4.44e6i)T + 1.22e16iT^{2} \)
31 \( 1 + 1.41e8T + 2.54e16T^{2} \)
37 \( 1 + (-4.51e7 + 4.51e7i)T - 1.77e17iT^{2} \)
41 \( 1 + 3.91e8iT - 5.50e17T^{2} \)
43 \( 1 + (-8.76e8 + 8.76e8i)T - 9.29e17iT^{2} \)
47 \( 1 + 1.57e9T + 2.47e18T^{2} \)
53 \( 1 + (3.45e9 - 3.45e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (5.84e9 - 5.84e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-1.80e9 - 1.80e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (5.33e9 + 5.33e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 6.24e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.44e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.55e8T + 7.47e20T^{2} \)
83 \( 1 + (-5.06e8 - 5.06e8i)T + 1.28e21iT^{2} \)
89 \( 1 - 4.97e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.39e10T + 7.15e21T^{2} \)
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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

−16.92867955768199371601555215408, −15.81728216588505480126092896800, −13.77240293951549055803548918074, −12.76178418346802473433755260392, −11.96940256679530788009332124082, −9.296917396431104193470013751289, −7.56348636299946189714019462971, −5.84434742708443330966886422070, −5.35737227928772801546051256425, −1.57055331308372955021749335559, 0.51840656566161467837431182763, 3.16275015046054309938154413337, 4.87731603301900207555436552154, 6.21207639536678586924692750616, 9.685875719518262677577454674315, 10.67140959627533002234602814880, 11.12363738691216640393909946544, 13.25058585448565315962464991347, 14.45710182294310854393187249513, 16.06432009358966742173861150998

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