Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.980 + 0.194i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−26.9 + 36.3i)2-s + (−199. + 199. i)3-s + (−593. − 1.96e3i)4-s + (8.93e3 + 8.93e3i)5-s + (−1.87e3 − 1.26e4i)6-s + 5.56e4i·7-s + (8.72e4 + 3.12e4i)8-s + 9.75e4i·9-s + (−5.65e5 + 8.37e4i)10-s + (−1.46e5 − 1.46e5i)11-s + (5.09e5 + 2.72e5i)12-s + (1.33e6 − 1.33e6i)13-s + (−2.02e6 − 1.50e6i)14-s − 3.56e6·15-s + (−3.48e6 + 2.32e6i)16-s − 2.92e6·17-s + ⋯
L(s)  = 1  + (−0.595 + 0.803i)2-s + (−0.474 + 0.474i)3-s + (−0.289 − 0.957i)4-s + (1.27 + 1.27i)5-s + (−0.0981 − 0.663i)6-s + 1.25i·7-s + (0.941 + 0.337i)8-s + 0.550i·9-s + (−1.78 + 0.264i)10-s + (−0.273 − 0.273i)11-s + (0.591 + 0.316i)12-s + (0.994 − 0.994i)13-s + (−1.00 − 0.745i)14-s − 1.21·15-s + (−0.832 + 0.554i)16-s − 0.499·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.980 + 0.194i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.980 + 0.194i)$
$L(6)$  $\approx$  $0.109957 - 1.12250i$
$L(\frac12)$  $\approx$  $0.109957 - 1.12250i$
$L(\frac{13}{2})$   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 \( 1 + (26.9 - 36.3i)T \)
good3 \( 1 + (199. - 199. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-8.93e3 - 8.93e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 5.56e4iT - 1.97e9T^{2} \)
11 \( 1 + (1.46e5 + 1.46e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-1.33e6 + 1.33e6i)T - 1.79e12iT^{2} \)
17 \( 1 + 2.92e6T + 3.42e13T^{2} \)
19 \( 1 + (1.11e7 - 1.11e7i)T - 1.16e14iT^{2} \)
23 \( 1 + 2.12e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.31e7 - 1.31e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 1.84e8T + 2.54e16T^{2} \)
37 \( 1 + (-6.58e7 - 6.58e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 7.48e8iT - 5.50e17T^{2} \)
43 \( 1 + (3.10e8 + 3.10e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.85e9T + 2.47e18T^{2} \)
53 \( 1 + (-3.45e9 - 3.45e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-8.57e8 - 8.57e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (-6.80e9 + 6.80e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (5.85e9 - 5.85e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.15e8iT - 2.31e20T^{2} \)
73 \( 1 - 1.23e9iT - 3.13e20T^{2} \)
79 \( 1 + 1.66e9T + 7.47e20T^{2} \)
83 \( 1 + (1.21e10 - 1.21e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 3.35e10iT - 2.77e21T^{2} \)
97 \( 1 - 2.49e10T + 7.15e21T^{2} \)
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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

−17.21977968967705171547830781242, −15.77962805930527146146911283188, −14.79687552860518145418085774319, −13.46062857239805512015713823676, −10.84899234124466745150985659487, −10.13608292370569827369649557900, −8.458204969450209279967575357039, −6.28124262933685264477616231978, −5.54679912060190375039367709395, −2.24711918514876413932432083748, 0.65156579326706514811892427125, 1.66634061721645076911469364080, 4.40815308794584880158225682877, 6.62885794542260829394748591172, 8.730158078871313417531896623248, 9.900596218747485098870834403960, 11.43558772230299615417117520094, 13.04099803846429684611330023034, 13.48997708905531510437442251073, 16.46555246770178549426922455633

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