Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.824 - 0.566i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (33.6 − 30.2i)2-s + (−145. + 145. i)3-s + (214. − 2.03e3i)4-s + (−4.65e3 − 4.65e3i)5-s + (−486. + 9.27e3i)6-s + 4.32e4i·7-s + (−5.44e4 − 7.49e4i)8-s + 1.35e5i·9-s + (−2.97e5 − 1.56e4i)10-s + (−6.85e5 − 6.85e5i)11-s + (2.64e5 + 3.26e5i)12-s + (−1.40e6 + 1.40e6i)13-s + (1.30e6 + 1.45e6i)14-s + 1.34e6·15-s + (−4.10e6 − 8.72e5i)16-s + 3.84e6·17-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.344 + 0.344i)3-s + (0.104 − 0.994i)4-s + (−0.665 − 0.665i)5-s + (−0.0255 + 0.486i)6-s + 0.972i·7-s + (−0.587 − 0.809i)8-s + 0.762i·9-s + (−0.940 − 0.0493i)10-s + (−1.28 − 1.28i)11-s + (0.306 + 0.378i)12-s + (−1.04 + 1.04i)13-s + (0.650 + 0.722i)14-s + 0.459·15-s + (−0.978 − 0.208i)16-s + 0.656·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(12-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.824 - 0.566i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.824 - 0.566i)$
$L(6)$  $\approx$  $0.0703914 + 0.226609i$
$L(\frac12)$  $\approx$  $0.0703914 + 0.226609i$
$L(\frac{13}{2})$   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 \( 1 + (-33.6 + 30.2i)T \)
good3 \( 1 + (145. - 145. i)T - 1.77e5iT^{2} \)
5 \( 1 + (4.65e3 + 4.65e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 4.32e4iT - 1.97e9T^{2} \)
11 \( 1 + (6.85e5 + 6.85e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (1.40e6 - 1.40e6i)T - 1.79e12iT^{2} \)
17 \( 1 - 3.84e6T + 3.42e13T^{2} \)
19 \( 1 + (7.81e6 - 7.81e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 1.99e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.18e8 + 1.18e8i)T - 1.22e16iT^{2} \)
31 \( 1 - 5.32e7T + 2.54e16T^{2} \)
37 \( 1 + (3.70e8 + 3.70e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 8.83e7iT - 5.50e17T^{2} \)
43 \( 1 + (-1.81e8 - 1.81e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.92e9T + 2.47e18T^{2} \)
53 \( 1 + (-1.39e9 - 1.39e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (2.09e9 + 2.09e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-1.23e9 + 1.23e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (9.73e9 - 9.73e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 2.30e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.18e10iT - 3.13e20T^{2} \)
79 \( 1 + 1.21e8T + 7.47e20T^{2} \)
83 \( 1 + (4.69e10 - 4.69e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 6.68e10iT - 2.77e21T^{2} \)
97 \( 1 - 4.52e10T + 7.15e21T^{2} \)
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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

−15.67268087326054631945010255511, −14.10599558606606280572981317443, −12.58133161376158491090633196019, −11.61798240605425545026035708221, −10.25262502401925425436777354401, −8.368886730519757743827145059689, −5.68142834532944676920640840084, −4.53502952500212921927537633659, −2.46549648243321155017067791248, −0.07886282812753708948790843771, 3.14285079519994646074491927167, 4.92131942693315712579191969088, 6.94839516997619059053348053392, 7.66396031003038033503671126897, 10.38547013057894201964154252024, 12.08506220011877270880267259037, 13.08361679003986409935147727489, 14.79509464387958969476543859452, 15.49592157649862714566166386161, 17.24310743411184464809911354314

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