Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.824 + 0.565i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (30.2 − 33.6i)2-s + (−56.4 − 56.4i)3-s + (−213. − 2.03e3i)4-s + (6.26e3 − 6.26e3i)5-s + (−3.60e3 + 188. i)6-s + 3.29e4i·7-s + (−7.49e4 − 5.45e4i)8-s − 1.70e5i·9-s + (−2.08e4 − 4.00e5i)10-s + (−1.51e5 + 1.51e5i)11-s + (−1.02e5 + 1.27e5i)12-s + (−1.05e6 − 1.05e6i)13-s + (1.10e6 + 9.97e5i)14-s − 7.06e5·15-s + (−4.10e6 + 8.68e5i)16-s − 1.95e6·17-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.134 − 0.134i)3-s + (−0.104 − 0.994i)4-s + (0.895 − 0.895i)5-s + (−0.189 + 0.00989i)6-s + 0.740i·7-s + (−0.808 − 0.588i)8-s − 0.964i·9-s + (−0.0660 − 1.26i)10-s + (−0.284 + 0.284i)11-s + (−0.119 + 0.147i)12-s + (−0.785 − 0.785i)13-s + (0.550 + 0.495i)14-s − 0.240·15-s + (−0.978 + 0.207i)16-s − 0.334·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(12-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.824 + 0.565i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.824 + 0.565i)$
$L(6)$  $\approx$  $0.702127 - 2.26452i$
$L(\frac12)$  $\approx$  $0.702127 - 2.26452i$
$L(\frac{13}{2})$   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 \( 1 + (-30.2 + 33.6i)T \)
good3 \( 1 + (56.4 + 56.4i)T + 1.77e5iT^{2} \)
5 \( 1 + (-6.26e3 + 6.26e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 3.29e4iT - 1.97e9T^{2} \)
11 \( 1 + (1.51e5 - 1.51e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (1.05e6 + 1.05e6i)T + 1.79e12iT^{2} \)
17 \( 1 + 1.95e6T + 3.42e13T^{2} \)
19 \( 1 + (-5.25e6 - 5.25e6i)T + 1.16e14iT^{2} \)
23 \( 1 + 4.62e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.16e8 - 1.16e8i)T + 1.22e16iT^{2} \)
31 \( 1 - 2.86e8T + 2.54e16T^{2} \)
37 \( 1 + (-3.33e8 + 3.33e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.31e8iT - 5.50e17T^{2} \)
43 \( 1 + (-5.60e8 + 5.60e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.10e8T + 2.47e18T^{2} \)
53 \( 1 + (2.81e9 - 2.81e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (3.57e8 - 3.57e8i)T - 3.01e19iT^{2} \)
61 \( 1 + (6.15e9 + 6.15e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-8.27e9 - 8.27e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.51e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.11e10iT - 3.13e20T^{2} \)
79 \( 1 - 1.68e10T + 7.47e20T^{2} \)
83 \( 1 + (1.04e10 + 1.04e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 3.00e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.53e11T + 7.15e21T^{2} \)
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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

−15.58867489506273503127278396368, −14.28895384418061411659277801545, −12.72980723351335780324562555091, −12.20223426197004716339906600724, −10.17685745828792393850654287700, −8.994625059793058829556442490552, −6.11651922967682370317302346374, −4.87823199938944755103762045527, −2.57628475639247273336746855942, −0.863145047600622248615643477323, 2.62532035597697370477153218971, 4.71807208116558157715365899006, 6.35779715461784010542290111747, 7.66953050101767032675912518004, 9.899656554768541912458828383637, 11.44115228987899609690418394531, 13.57891630336642970499523603010, 13.95678986839241755029891873155, 15.55172768845498938387064723335, 16.89915904015531824219329146431

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