Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.730 - 0.682i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 + 0.253i)2-s + (5.49 + 5.49i)3-s + (7.87 − 1.42i)4-s + (−4.66 + 4.66i)5-s + (−16.8 − 14.0i)6-s − 24.8i·7-s + (−21.8 + 6.00i)8-s + 33.4i·9-s + (11.9 − 14.3i)10-s + (22.3 − 22.3i)11-s + (51.1 + 35.4i)12-s + (−11.2 − 11.2i)13-s + (6.30 + 70.1i)14-s − 51.2·15-s + (59.9 − 22.4i)16-s − 88.4·17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0894i)2-s + (1.05 + 1.05i)3-s + (0.983 − 0.178i)4-s + (−0.417 + 0.417i)5-s + (−1.14 − 0.958i)6-s − 1.34i·7-s + (−0.964 + 0.265i)8-s + 1.23i·9-s + (0.378 − 0.452i)10-s + (0.612 − 0.612i)11-s + (1.22 + 0.852i)12-s + (−0.240 − 0.240i)13-s + (0.120 + 1.33i)14-s − 0.882·15-s + (0.936 − 0.350i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.730 - 0.682i$
motivic weight  =  \(3\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :3/2),\ 0.730 - 0.682i)$
$L(2)$  $\approx$  $0.799302 + 0.315141i$
$L(\frac12)$  $\approx$  $0.799302 + 0.315141i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (2.81 - 0.253i)T \)
good3 \( 1 + (-5.49 - 5.49i)T + 27iT^{2} \)
5 \( 1 + (4.66 - 4.66i)T - 125iT^{2} \)
7 \( 1 + 24.8iT - 343T^{2} \)
11 \( 1 + (-22.3 + 22.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (11.2 + 11.2i)T + 2.19e3iT^{2} \)
17 \( 1 + 88.4T + 4.91e3T^{2} \)
19 \( 1 + (-37.8 - 37.8i)T + 6.85e3iT^{2} \)
23 \( 1 - 48.1iT - 1.21e4T^{2} \)
29 \( 1 + (-10.4 - 10.4i)T + 2.43e4iT^{2} \)
31 \( 1 + 96.9T + 2.97e4T^{2} \)
37 \( 1 + (163. - 163. i)T - 5.06e4iT^{2} \)
41 \( 1 + 360. iT - 6.89e4T^{2} \)
43 \( 1 + (100. - 100. i)T - 7.95e4iT^{2} \)
47 \( 1 - 220.T + 1.03e5T^{2} \)
53 \( 1 + (175. - 175. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-405. + 405. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-664. - 664. i)T + 2.26e5iT^{2} \)
67 \( 1 + (107. + 107. i)T + 3.00e5iT^{2} \)
71 \( 1 - 215. iT - 3.57e5T^{2} \)
73 \( 1 + 668. iT - 3.89e5T^{2} \)
79 \( 1 + 822.T + 4.93e5T^{2} \)
83 \( 1 + (-326. - 326. i)T + 5.71e5iT^{2} \)
89 \( 1 + 262. iT - 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−19.18378111044967489493928575706, −17.37080965920219412077854324230, −16.16604314025702061232247799118, −15.10379471695498216268016039356, −13.92801851059991218948762916483, −11.16800947255393671871768887243, −10.08574417320134893822762914855, −8.756413192025216190940882303818, −7.25148595476539696283598629308, −3.60641650265063205159219435507, 2.25265769513463768454853155610, 6.86678798802957610728685683430, 8.395618962321159118455322537640, 9.208322658304332383728951439167, 11.75882876176833103648751854754, 12.74861254480874746755156540661, 14.69988507753987762872501130772, 15.86252342172390217843419830064, 17.70774264941209122571400972689, 18.64532290781138716466997533540

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