Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.864 - 0.502i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.51 − 45.2i)2-s + (442. + 442. i)3-s + (−2.04e3 + 137. i)4-s + (2.49e3 − 2.49e3i)5-s + (1.93e4 − 2.06e4i)6-s + 3.93e4i·7-s + (9.29e3 + 9.22e4i)8-s + 2.13e5i·9-s + (−1.16e5 − 1.09e5i)10-s + (−3.11e5 + 3.11e5i)11-s + (−9.63e5 − 8.42e5i)12-s + (1.47e6 + 1.47e6i)13-s + (1.78e6 − 5.96e4i)14-s + 2.20e6·15-s + (4.15e6 − 5.59e5i)16-s + 6.43e6·17-s + ⋯
L(s)  = 1  + (−0.0334 − 0.999i)2-s + (1.05 + 1.05i)3-s + (−0.997 + 0.0669i)4-s + (0.357 − 0.357i)5-s + (1.01 − 1.08i)6-s + 0.885i·7-s + (0.100 + 0.994i)8-s + 1.20i·9-s + (−0.369 − 0.345i)10-s + (−0.582 + 0.582i)11-s + (−1.11 − 0.977i)12-s + (1.09 + 1.09i)13-s + (0.885 − 0.0296i)14-s + 0.750·15-s + (0.991 − 0.133i)16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.864 - 0.502i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.864 - 0.502i)$
$L(6)$  $\approx$  $2.18955 + 0.590219i$
$L(\frac12)$  $\approx$  $2.18955 + 0.590219i$
$L(\frac{13}{2})$   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 \( 1 + (1.51 + 45.2i)T \)
good3 \( 1 + (-442. - 442. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-2.49e3 + 2.49e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 3.93e4iT - 1.97e9T^{2} \)
11 \( 1 + (3.11e5 - 3.11e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-1.47e6 - 1.47e6i)T + 1.79e12iT^{2} \)
17 \( 1 - 6.43e6T + 3.42e13T^{2} \)
19 \( 1 + (3.94e6 + 3.94e6i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.40e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.10e8 - 1.10e8i)T + 1.22e16iT^{2} \)
31 \( 1 + 2.26e8T + 2.54e16T^{2} \)
37 \( 1 + (2.38e8 - 2.38e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.45e9iT - 5.50e17T^{2} \)
43 \( 1 + (-1.31e8 + 1.31e8i)T - 9.29e17iT^{2} \)
47 \( 1 + 2.03e9T + 2.47e18T^{2} \)
53 \( 1 + (-1.22e9 + 1.22e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-2.81e9 + 2.81e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (5.47e8 + 5.47e8i)T + 4.35e19iT^{2} \)
67 \( 1 + (1.17e10 + 1.17e10i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.29e9iT - 2.31e20T^{2} \)
73 \( 1 + 2.25e10iT - 3.13e20T^{2} \)
79 \( 1 - 3.10e10T + 7.47e20T^{2} \)
83 \( 1 + (-3.09e10 - 3.09e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 7.63e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.73e10T + 7.15e21T^{2} \)
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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

−16.37767652062064752554193624970, −14.95541541550471151392507179554, −13.82285174341065254448938528186, −12.37734169300329457532346288984, −10.61136037023208512005376908432, −9.309691452161463005817642913823, −8.605164399023146620329956574157, −5.04718844172274439538202777921, −3.49246819560522740160869180841, −1.97907023357941058952542357349, 0.981528098618663919581607074173, 3.35147774459096383328329871437, 6.01136501316007081243441204743, 7.54300252606032821000408016596, 8.352436270316630947819987387557, 10.24541172347438059176283000629, 13.01187451771015635928875212467, 13.69985093421602064895889817045, 14.66611667923655505238998735529, 16.23236974521010436726859380047

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