Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.923 + 0.382i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (45.2 + 0.00312i)2-s + (457. − 457. i)3-s + (2.04e3 + 0.283i)4-s + (5.57e3 + 5.57e3i)5-s + (2.06e4 − 2.06e4i)6-s + 5.54e4i·7-s + (9.26e4 + 19.2i)8-s − 2.41e5i·9-s + (2.52e5 + 2.52e5i)10-s + (−5.93e5 − 5.93e5i)11-s + (9.36e5 − 9.36e5i)12-s + (−3.20e5 + 3.20e5i)13-s + (−173. + 2.50e6i)14-s + 5.09e6·15-s + (4.19e6 + 1.15e3i)16-s − 9.66e6·17-s + ⋯
L(s)  = 1  + (0.999 + 6.91e−5i)2-s + (1.08 − 1.08i)3-s + (0.999 + 0.000138i)4-s + (0.797 + 0.797i)5-s + (1.08 − 1.08i)6-s + 1.24i·7-s + (0.999 + 0.000207i)8-s − 1.36i·9-s + (0.797 + 0.797i)10-s + (−1.11 − 1.11i)11-s + (1.08 − 1.08i)12-s + (−0.239 + 0.239i)13-s + (−8.61e−5 + 1.24i)14-s + 1.73·15-s + (0.999 + 0.000276i)16-s − 1.65·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.923 + 0.382i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.923 + 0.382i)$
$L(6)$  $\approx$  $4.70025 - 0.935614i$
$L(\frac12)$  $\approx$  $4.70025 - 0.935614i$
$L(\frac{13}{2})$   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 \( 1 + (-45.2 - 0.00312i)T \)
good3 \( 1 + (-457. + 457. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-5.57e3 - 5.57e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 5.54e4iT - 1.97e9T^{2} \)
11 \( 1 + (5.93e5 + 5.93e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (3.20e5 - 3.20e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 9.66e6T + 3.42e13T^{2} \)
19 \( 1 + (-6.35e6 + 6.35e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 1.96e7iT - 9.52e14T^{2} \)
29 \( 1 + (7.64e7 - 7.64e7i)T - 1.22e16iT^{2} \)
31 \( 1 + 1.66e8T + 2.54e16T^{2} \)
37 \( 1 + (-2.74e7 - 2.74e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 6.15e8iT - 5.50e17T^{2} \)
43 \( 1 + (-8.26e8 - 8.26e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.20e9T + 2.47e18T^{2} \)
53 \( 1 + (-6.02e8 - 6.02e8i)T + 9.26e18iT^{2} \)
59 \( 1 + (-3.88e9 - 3.88e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (2.43e9 - 2.43e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-8.86e9 + 8.86e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 2.93e10iT - 2.31e20T^{2} \)
73 \( 1 - 7.45e9iT - 3.13e20T^{2} \)
79 \( 1 + 2.31e10T + 7.47e20T^{2} \)
83 \( 1 + (-4.40e9 + 4.40e9i)T - 1.28e21iT^{2} \)
89 \( 1 + 4.82e10iT - 2.77e21T^{2} \)
97 \( 1 - 9.19e10T + 7.15e21T^{2} \)
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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

−15.74626548694266179498369577945, −14.55533029029004500322331926408, −13.61594565386915900074588199103, −12.76572215655985538836396114088, −11.04513534212228964948512994865, −8.763115684235990513107534334725, −7.07768633878631956988705472012, −5.74686195744362160037851263433, −2.77370150150519398220464734163, −2.24024601012408360576151151533, 2.14604163173384430837558248370, 3.96799578674276362647210661678, 5.07632029742480879335587651658, 7.56345081231120927586400159316, 9.553779530916189331285143996601, 10.61591352450556858829541336026, 13.01910347719696834961160376700, 13.73332684226616422995732580876, 15.04547101716886721781245903508, 16.04254605592016213227302195694

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