Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.625 - 0.780i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (40.4 − 20.2i)2-s + (−522. − 522. i)3-s + (1.22e3 − 1.63e3i)4-s + (−3.20e3 + 3.20e3i)5-s + (−3.17e4 − 1.05e4i)6-s − 3.19e4i·7-s + (1.66e4 − 9.11e4i)8-s + 3.69e5i·9-s + (−6.48e4 + 1.94e5i)10-s + (−4.06e5 + 4.06e5i)11-s + (−1.49e6 + 2.13e5i)12-s + (1.12e6 + 1.12e6i)13-s + (−6.46e5 − 1.29e6i)14-s + 3.34e6·15-s + (−1.16e6 − 4.02e6i)16-s − 5.55e6·17-s + ⋯
L(s)  = 1  + (0.894 − 0.446i)2-s + (−1.24 − 1.24i)3-s + (0.600 − 0.799i)4-s + (−0.458 + 0.458i)5-s + (−1.66 − 0.555i)6-s − 0.719i·7-s + (0.179 − 0.983i)8-s + 2.08i·9-s + (−0.205 + 0.614i)10-s + (−0.761 + 0.761i)11-s + (−1.73 + 0.247i)12-s + (0.840 + 0.840i)13-s + (−0.321 − 0.643i)14-s + 1.13·15-s + (−0.278 − 0.960i)16-s − 0.948·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.625 - 0.780i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.625 - 0.780i)$
$L(6)$  $\approx$  $0.299097 + 0.622679i$
$L(\frac12)$  $\approx$  $0.299097 + 0.622679i$
$L(\frac{13}{2})$   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 \( 1 + (-40.4 + 20.2i)T \)
good3 \( 1 + (522. + 522. i)T + 1.77e5iT^{2} \)
5 \( 1 + (3.20e3 - 3.20e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 3.19e4iT - 1.97e9T^{2} \)
11 \( 1 + (4.06e5 - 4.06e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-1.12e6 - 1.12e6i)T + 1.79e12iT^{2} \)
17 \( 1 + 5.55e6T + 3.42e13T^{2} \)
19 \( 1 + (7.81e6 + 7.81e6i)T + 1.16e14iT^{2} \)
23 \( 1 + 3.53e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.32e8 + 1.32e8i)T + 1.22e16iT^{2} \)
31 \( 1 - 7.64e7T + 2.54e16T^{2} \)
37 \( 1 + (6.43e7 - 6.43e7i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.19e9iT - 5.50e17T^{2} \)
43 \( 1 + (1.26e9 - 1.26e9i)T - 9.29e17iT^{2} \)
47 \( 1 - 5.29e8T + 2.47e18T^{2} \)
53 \( 1 + (3.02e9 - 3.02e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-2.78e9 + 2.78e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (3.36e9 + 3.36e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (1.24e9 + 1.24e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.89e9iT - 2.31e20T^{2} \)
73 \( 1 + 7.84e9iT - 3.13e20T^{2} \)
79 \( 1 - 1.00e10T + 7.47e20T^{2} \)
83 \( 1 + (4.69e9 + 4.69e9i)T + 1.28e21iT^{2} \)
89 \( 1 + 2.92e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.08e10T + 7.15e21T^{2} \)
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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

−15.45478626076423773942573149361, −13.62649848090631497168492150796, −12.78767797211075657226989865138, −11.43520080043239445279734668197, −10.72648073589275823758940514796, −7.26147221886540176025245889729, −6.34811315770782807719612652966, −4.51486342990894471158214207614, −1.98292070113385797450143109149, −0.24899681324705062947198842736, 3.62932059869859445766483912460, 5.10249510115381750173586010518, 6.01699956574689765811053891748, 8.506539357554024871923182387725, 10.71849376831275474240863856861, 11.70936377463650912184986802579, 13.02228427075084664446014131772, 15.19415390417306117818754395930, 15.83206156226570299145566225386, 16.67267737563158084044541003982

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