Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.507 + 0.861i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−42.3 − 15.8i)2-s + (−435. − 435. i)3-s + (1.54e3 + 1.34e3i)4-s + (−1.51e3 + 1.51e3i)5-s + (1.15e4 + 2.53e4i)6-s + 5.97e4i·7-s + (−4.41e4 − 8.14e4i)8-s + 2.01e5i·9-s + (8.83e4 − 4.02e4i)10-s + (5.61e4 − 5.61e4i)11-s + (−8.76e4 − 1.25e6i)12-s + (−3.15e5 − 3.15e5i)13-s + (9.46e5 − 2.53e6i)14-s + 1.32e6·15-s + (5.81e5 + 4.15e6i)16-s + 8.11e6·17-s + ⋯
L(s)  = 1  + (−0.936 − 0.350i)2-s + (−1.03 − 1.03i)3-s + (0.754 + 0.656i)4-s + (−0.217 + 0.217i)5-s + (0.606 + 1.33i)6-s + 1.34i·7-s + (−0.476 − 0.878i)8-s + 1.13i·9-s + (0.279 − 0.127i)10-s + (0.105 − 0.105i)11-s + (−0.101 − 1.45i)12-s + (−0.235 − 0.235i)13-s + (0.470 − 1.25i)14-s + 0.448·15-s + (0.138 + 0.990i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.507 + 0.861i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.507 + 0.861i)$
$L(6)$  $\approx$  $0.576376 - 0.329594i$
$L(\frac12)$  $\approx$  $0.576376 - 0.329594i$
$L(\frac{13}{2})$   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 \( 1 + (42.3 + 15.8i)T \)
good3 \( 1 + (435. + 435. i)T + 1.77e5iT^{2} \)
5 \( 1 + (1.51e3 - 1.51e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 5.97e4iT - 1.97e9T^{2} \)
11 \( 1 + (-5.61e4 + 5.61e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (3.15e5 + 3.15e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 8.11e6T + 3.42e13T^{2} \)
19 \( 1 + (1.33e7 + 1.33e7i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.84e7iT - 9.52e14T^{2} \)
29 \( 1 + (-7.59e7 - 7.59e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 5.67e7T + 2.54e16T^{2} \)
37 \( 1 + (-3.75e8 + 3.75e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.51e7iT - 5.50e17T^{2} \)
43 \( 1 + (-1.08e9 + 1.08e9i)T - 9.29e17iT^{2} \)
47 \( 1 - 2.79e9T + 2.47e18T^{2} \)
53 \( 1 + (-2.03e9 + 2.03e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (4.23e9 - 4.23e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-6.54e9 - 6.54e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (8.01e9 + 8.01e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.40e10iT - 2.31e20T^{2} \)
73 \( 1 - 6.29e9iT - 3.13e20T^{2} \)
79 \( 1 + 3.46e10T + 7.47e20T^{2} \)
83 \( 1 + (-4.29e10 - 4.29e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 3.39e9iT - 2.77e21T^{2} \)
97 \( 1 - 1.55e10T + 7.15e21T^{2} \)
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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

−16.69209099444655347592738552147, −15.18980364928030474605592148143, −12.65534531601849409314178083055, −11.99865748073508989227005977000, −10.78586427393216213422744232876, −8.828763326167307310440062267303, −7.23736960537878785360380230454, −5.87209144485914852256116037072, −2.44964136545566845574839395045, −0.69392166129695762388087347077, 0.827201437368612026926467284042, 4.28849254666338679475054300294, 6.04767627198396589670548690927, 7.80772442862509634316337693137, 9.881676226220484634273971835459, 10.55322910623258569152471309492, 11.90937406764523498854847165934, 14.40093286763252144862333705885, 15.88412105553498312559363757104, 16.82151203595870565514827318347

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