Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.922 + 0.385i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (25.1 − 37.6i)2-s + (367. − 367. i)3-s + (−786. − 1.89e3i)4-s + (−502. − 502. i)5-s + (−4.60e3 − 2.30e4i)6-s − 3.30e4i·7-s + (−9.09e4 − 1.78e4i)8-s − 9.34e4i·9-s + (−3.15e4 + 6.29e3i)10-s + (3.35e5 + 3.35e5i)11-s + (−9.84e5 − 4.06e5i)12-s + (5.68e5 − 5.68e5i)13-s + (−1.24e6 − 8.30e5i)14-s − 3.69e5·15-s + (−2.95e6 + 2.97e6i)16-s − 6.78e6·17-s + ⋯
L(s)  = 1  + (0.554 − 0.831i)2-s + (0.873 − 0.873i)3-s + (−0.384 − 0.923i)4-s + (−0.0718 − 0.0718i)5-s + (−0.242 − 1.21i)6-s − 0.743i·7-s + (−0.981 − 0.192i)8-s − 0.527i·9-s + (−0.0996 + 0.0199i)10-s + (0.628 + 0.628i)11-s + (−1.14 − 0.471i)12-s + (0.424 − 0.424i)13-s + (−0.618 − 0.412i)14-s − 0.125·15-s + (−0.705 + 0.709i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.922 + 0.385i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.922 + 0.385i)$
$L(6)$  $\approx$  $0.552672 - 2.75741i$
$L(\frac12)$  $\approx$  $0.552672 - 2.75741i$
$L(\frac{13}{2})$   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 \( 1 + (-25.1 + 37.6i)T \)
good3 \( 1 + (-367. + 367. i)T - 1.77e5iT^{2} \)
5 \( 1 + (502. + 502. i)T + 4.88e7iT^{2} \)
7 \( 1 + 3.30e4iT - 1.97e9T^{2} \)
11 \( 1 + (-3.35e5 - 3.35e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-5.68e5 + 5.68e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 6.78e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.33e6 + 1.33e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 1.07e6iT - 9.52e14T^{2} \)
29 \( 1 + (-5.01e7 + 5.01e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 2.95e8T + 2.54e16T^{2} \)
37 \( 1 + (-5.11e8 - 5.11e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 9.31e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.27e9 + 1.27e9i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.80e7T + 2.47e18T^{2} \)
53 \( 1 + (-2.29e9 - 2.29e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (4.81e8 + 4.81e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (-1.30e9 + 1.30e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (9.42e9 - 9.42e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.74e10iT - 2.31e20T^{2} \)
73 \( 1 - 8.91e9iT - 3.13e20T^{2} \)
79 \( 1 + 4.07e10T + 7.47e20T^{2} \)
83 \( 1 + (-1.16e10 + 1.16e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 7.96e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.03e10T + 7.15e21T^{2} \)
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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

−15.32250040371244216659540488195, −13.92796632851482633888752868055, −13.28457480889426226732362536156, −11.91445579662719424269760593354, −10.24570900191103338683942635855, −8.533218258466162544240278594415, −6.72322196210743402121499367593, −4.27746914593437721887234692689, −2.50552250421454851537090672594, −1.01116125751642421646661300701, 3.03245862408555192145280788201, 4.42524023940066340716052647510, 6.32814746317604027001678269915, 8.419747374786243924724343321922, 9.298688872351037046883913064799, 11.61036722751807476295136775063, 13.40566894976494179595265139739, 14.60067662819781808152178998466, 15.45270734957256090342744789361, 16.45737576660152868730717858135

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