Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.977 + 0.211i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−36.4 − 26.8i)2-s + (−475. + 475. i)3-s + (610. + 1.95e3i)4-s + (2.80e3 + 2.80e3i)5-s + (3.01e4 − 4.59e3i)6-s + 7.60e4i·7-s + (3.01e4 − 8.76e4i)8-s − 2.75e5i·9-s + (−2.70e4 − 1.77e5i)10-s + (4.09e5 + 4.09e5i)11-s + (−1.22e6 − 6.39e5i)12-s + (−1.80e6 + 1.80e6i)13-s + (2.03e6 − 2.77e6i)14-s − 2.67e6·15-s + (−3.44e6 + 2.38e6i)16-s − 1.50e6·17-s + ⋯
L(s)  = 1  + (−0.805 − 0.592i)2-s + (−1.13 + 1.13i)3-s + (0.298 + 0.954i)4-s + (0.401 + 0.401i)5-s + (1.58 − 0.241i)6-s + 1.70i·7-s + (0.325 − 0.945i)8-s − 1.55i·9-s + (−0.0856 − 0.561i)10-s + (0.767 + 0.767i)11-s + (−1.41 − 0.742i)12-s + (−1.34 + 1.34i)13-s + (1.01 − 1.37i)14-s − 0.907·15-s + (−0.822 + 0.569i)16-s − 0.257·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(12-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.977 + 0.211i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.977 + 0.211i)$
$L(6)$  $\approx$  $0.0574120 - 0.537442i$
$L(\frac12)$  $\approx$  $0.0574120 - 0.537442i$
$L(\frac{13}{2})$   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 \( 1 + (36.4 + 26.8i)T \)
good3 \( 1 + (475. - 475. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-2.80e3 - 2.80e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 7.60e4iT - 1.97e9T^{2} \)
11 \( 1 + (-4.09e5 - 4.09e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (1.80e6 - 1.80e6i)T - 1.79e12iT^{2} \)
17 \( 1 + 1.50e6T + 3.42e13T^{2} \)
19 \( 1 + (-5.45e6 + 5.45e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 2.28e6iT - 9.52e14T^{2} \)
29 \( 1 + (-4.38e7 + 4.38e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 2.76e7T + 2.54e16T^{2} \)
37 \( 1 + (-2.40e8 - 2.40e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 1.08e8iT - 5.50e17T^{2} \)
43 \( 1 + (-1.91e8 - 1.91e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.72e9T + 2.47e18T^{2} \)
53 \( 1 + (8.00e7 + 8.00e7i)T + 9.26e18iT^{2} \)
59 \( 1 + (-6.70e9 - 6.70e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-9.82e7 + 9.82e7i)T - 4.35e19iT^{2} \)
67 \( 1 + (-6.92e9 + 6.92e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.20e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.00e10iT - 3.13e20T^{2} \)
79 \( 1 + 1.65e10T + 7.47e20T^{2} \)
83 \( 1 + (-2.19e10 + 2.19e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 3.11e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.12e9T + 7.15e21T^{2} \)
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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

−17.26607966975362351372110868625, −16.11389754294270708455843553023, −14.90724820492393226511100224341, −12.08642388610681648247074340590, −11.59031980970150623287850995325, −9.897267610867766652672962081997, −9.220824148973190124011392546589, −6.51663926697468375234917350892, −4.65965740607356029208393653174, −2.31480598586036920486554964640, 0.40195874175706597785959418791, 1.20262085526861008207412444628, 5.35879920151058858872709998089, 6.77473723973890520267962704141, 7.78309675236251760812770938912, 10.03027555040143812085519600626, 11.27000732313250356815578449649, 12.96061883975486308706455032506, 14.24207702201509748812870975468, 16.49472197970787008278510620163

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