Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.692 - 0.721i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−45.0 + 4.65i)2-s + (13.6 + 13.6i)3-s + (2.00e3 − 419. i)4-s + (−3.50e3 + 3.50e3i)5-s + (−675. − 548. i)6-s − 3.76e4i·7-s + (−8.82e4 + 2.82e4i)8-s − 1.76e5i·9-s + (1.41e5 − 1.73e5i)10-s + (−2.83e5 + 2.83e5i)11-s + (3.29e4 + 2.15e4i)12-s + (1.70e6 + 1.70e6i)13-s + (1.75e5 + 1.69e6i)14-s − 9.52e4·15-s + (3.84e6 − 1.68e6i)16-s + 1.57e6·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.102i)2-s + (0.0323 + 0.0323i)3-s + (0.978 − 0.204i)4-s + (−0.501 + 0.501i)5-s + (−0.0354 − 0.0288i)6-s − 0.846i·7-s + (−0.952 + 0.304i)8-s − 0.997i·9-s + (0.446 − 0.550i)10-s + (−0.530 + 0.530i)11-s + (0.0382 + 0.0250i)12-s + (1.27 + 1.27i)13-s + (0.0871 + 0.842i)14-s − 0.0323·15-s + (0.916 − 0.400i)16-s + 0.269·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(12-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.692 - 0.721i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.692 - 0.721i)$
$L(6)$  $\approx$  $0.926301 + 0.394528i$
$L(\frac12)$  $\approx$  $0.926301 + 0.394528i$
$L(\frac{13}{2})$   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 \( 1 + (45.0 - 4.65i)T \)
good3 \( 1 + (-13.6 - 13.6i)T + 1.77e5iT^{2} \)
5 \( 1 + (3.50e3 - 3.50e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 3.76e4iT - 1.97e9T^{2} \)
11 \( 1 + (2.83e5 - 2.83e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-1.70e6 - 1.70e6i)T + 1.79e12iT^{2} \)
17 \( 1 - 1.57e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.25e7 - 1.25e7i)T + 1.16e14iT^{2} \)
23 \( 1 + 6.83e6iT - 9.52e14T^{2} \)
29 \( 1 + (-5.00e7 - 5.00e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 7.19e7T + 2.54e16T^{2} \)
37 \( 1 + (-9.81e7 + 9.81e7i)T - 1.77e17iT^{2} \)
41 \( 1 - 7.24e8iT - 5.50e17T^{2} \)
43 \( 1 + (-1.93e8 + 1.93e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.39e9T + 2.47e18T^{2} \)
53 \( 1 + (3.32e9 - 3.32e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-5.46e9 + 5.46e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (4.88e9 + 4.88e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (7.47e9 + 7.47e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 2.58e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.88e10iT - 3.13e20T^{2} \)
79 \( 1 - 3.09e10T + 7.47e20T^{2} \)
83 \( 1 + (-2.96e10 - 2.96e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 6.42e10iT - 2.77e21T^{2} \)
97 \( 1 - 8.87e10T + 7.15e21T^{2} \)
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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

−16.62110543262935987678668949945, −15.54652941130837600732087621568, −14.18168335724811328675742240067, −12.00492132369150519462939398719, −10.77148656944959433888466635013, −9.410802810640770008749337777307, −7.72055924344267004138613431128, −6.48811404801039366944486019299, −3.55731008385668612819583218704, −1.16918475730803376399271951682, 0.74698789235937475480991847399, 2.81700371115127833264682695327, 5.60199133102614284451515429221, 7.83405715596612267775207153541, 8.726898236443444932842504665281, 10.53758261881596327212212039920, 11.76361298591856454115068467781, 13.28638817172062028974867766299, 15.64014768332296791717011625036, 16.05851319805227435329534394544

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