Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.479 + 0.877i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (26.1 − 36.9i)2-s + (255. + 255. i)3-s + (−681. − 1.93e3i)4-s + (−4.21e3 + 4.21e3i)5-s + (1.61e4 − 2.76e3i)6-s − 8.03e4i·7-s + (−8.91e4 − 2.53e4i)8-s − 4.61e4i·9-s + (4.55e4 + 2.65e5i)10-s + (6.11e5 − 6.11e5i)11-s + (3.19e5 − 6.68e5i)12-s + (3.70e5 + 3.70e5i)13-s + (−2.96e6 − 2.10e6i)14-s − 2.15e6·15-s + (−3.26e6 + 2.63e6i)16-s − 7.25e5·17-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)2-s + (0.608 + 0.608i)3-s + (−0.332 − 0.943i)4-s + (−0.603 + 0.603i)5-s + (0.847 − 0.145i)6-s − 1.80i·7-s + (−0.962 − 0.273i)8-s − 0.260i·9-s + (0.143 + 0.840i)10-s + (1.14 − 1.14i)11-s + (0.371 − 0.775i)12-s + (0.276 + 0.276i)13-s + (−1.47 − 1.04i)14-s − 0.733·15-s + (−0.778 + 0.627i)16-s − 0.123·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.479 + 0.877i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.479 + 0.877i)$
$L(6)$  $\approx$  $1.17979 - 1.98808i$
$L(\frac12)$  $\approx$  $1.17979 - 1.98808i$
$L(\frac{13}{2})$   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 \( 1 + (-26.1 + 36.9i)T \)
good3 \( 1 + (-255. - 255. i)T + 1.77e5iT^{2} \)
5 \( 1 + (4.21e3 - 4.21e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 8.03e4iT - 1.97e9T^{2} \)
11 \( 1 + (-6.11e5 + 6.11e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-3.70e5 - 3.70e5i)T + 1.79e12iT^{2} \)
17 \( 1 + 7.25e5T + 3.42e13T^{2} \)
19 \( 1 + (1.13e7 + 1.13e7i)T + 1.16e14iT^{2} \)
23 \( 1 - 4.89e7iT - 9.52e14T^{2} \)
29 \( 1 + (-4.98e7 - 4.98e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 4.20e7T + 2.54e16T^{2} \)
37 \( 1 + (1.05e8 - 1.05e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 4.49e8iT - 5.50e17T^{2} \)
43 \( 1 + (-7.89e8 + 7.89e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.32e9T + 2.47e18T^{2} \)
53 \( 1 + (-1.74e9 + 1.74e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (6.71e8 - 6.71e8i)T - 3.01e19iT^{2} \)
61 \( 1 + (8.30e8 + 8.30e8i)T + 4.35e19iT^{2} \)
67 \( 1 + (-3.88e9 - 3.88e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 5.14e9iT - 2.31e20T^{2} \)
73 \( 1 - 4.42e9iT - 3.13e20T^{2} \)
79 \( 1 - 2.71e10T + 7.47e20T^{2} \)
83 \( 1 + (-1.81e10 - 1.81e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 5.56e10iT - 2.77e21T^{2} \)
97 \( 1 - 2.61e10T + 7.15e21T^{2} \)
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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

−15.55586751334221486985720660579, −14.32211597569398384997272451105, −13.56725270759021088277929306558, −11.47231888927570028778616438488, −10.56895670046603763375635473994, −9.049785964032772444969918129191, −6.76991838174780158553835384101, −4.04894254031337630713953504733, −3.46523922444835153254618173959, −0.820884768682237316681174695910, 2.31996074780852540803052134706, 4.49620154337608936011067279184, 6.30307470655730899276489368010, 8.097026818645962859772334832208, 8.883945090996473560677713604247, 12.20776737883238778898568826828, 12.58301393647176931775904345293, 14.41975827228433298489690823779, 15.28499858415558438304635206660, 16.55240431816638840001165443731

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