Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.872 + 0.488i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (44.1 + 10.1i)2-s + (−179. − 179. i)3-s + (1.84e3 + 893. i)4-s + (2.96e3 − 2.96e3i)5-s + (−6.10e3 − 9.74e3i)6-s + 5.17e3i·7-s + (7.22e4 + 5.80e4i)8-s − 1.12e5i·9-s + (1.60e5 − 1.00e5i)10-s + (6.94e5 − 6.94e5i)11-s + (−1.70e5 − 4.91e5i)12-s + (5.24e5 + 5.24e5i)13-s + (−5.24e4 + 2.28e5i)14-s − 1.06e6·15-s + (2.59e6 + 3.29e6i)16-s + 1.82e6·17-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)2-s + (−0.426 − 0.426i)3-s + (0.899 + 0.436i)4-s + (0.423 − 0.423i)5-s + (−0.320 − 0.511i)6-s + 0.116i·7-s + (0.779 + 0.626i)8-s − 0.635i·9-s + (0.507 − 0.318i)10-s + (1.30 − 1.30i)11-s + (−0.197 − 0.570i)12-s + (0.391 + 0.391i)13-s + (−0.0260 + 0.113i)14-s − 0.361·15-s + (0.619 + 0.785i)16-s + 0.311·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.872 + 0.488i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.872 + 0.488i)$
$L(6)$  $\approx$  $3.10897 - 0.810444i$
$L(\frac12)$  $\approx$  $3.10897 - 0.810444i$
$L(\frac{13}{2})$   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 \( 1 + (-44.1 - 10.1i)T \)
good3 \( 1 + (179. + 179. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-2.96e3 + 2.96e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 5.17e3iT - 1.97e9T^{2} \)
11 \( 1 + (-6.94e5 + 6.94e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-5.24e5 - 5.24e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 1.82e6T + 3.42e13T^{2} \)
19 \( 1 + (6.01e5 + 6.01e5i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.32e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.89e5 - 1.89e5i)T + 1.22e16iT^{2} \)
31 \( 1 + 2.48e8T + 2.54e16T^{2} \)
37 \( 1 + (4.30e8 - 4.30e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.02e8iT - 5.50e17T^{2} \)
43 \( 1 + (6.91e7 - 6.91e7i)T - 9.29e17iT^{2} \)
47 \( 1 - 2.50e8T + 2.47e18T^{2} \)
53 \( 1 + (1.76e9 - 1.76e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-3.82e9 + 3.82e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-7.80e9 - 7.80e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (6.53e9 + 6.53e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.53e10iT - 2.31e20T^{2} \)
73 \( 1 - 2.49e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.46e10T + 7.47e20T^{2} \)
83 \( 1 + (2.00e10 + 2.00e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 3.72e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.06e11T + 7.15e21T^{2} \)
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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

−16.39220335066645136727858873675, −14.70765732249944076569668699746, −13.55782932424688301307352980706, −12.28676051218046271084662751391, −11.25129374691136028581780212816, −8.874153866925517457852308708749, −6.71387279782367111404163492986, −5.67079057196722250736352740971, −3.65156891535725530088129773672, −1.30268598160424434064689148834, 1.87963458465804296445417283273, 3.97370976762590022332261115610, 5.52064072406961491174766767805, 7.07352807825579483693343400274, 9.874128589725369867504143060788, 11.02590534056333130293647847893, 12.38922683895199978677399339343, 13.89187572120772259389180338147, 14.96513177382187483294332210992, 16.31184988488478538450011619063

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