Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.976 - 0.215i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.87 − 44.7i)2-s + (−291. + 291. i)3-s + (−1.95e3 + 614. i)4-s + (−9.75e3 − 9.75e3i)5-s + (1.50e4 + 1.10e4i)6-s − 1.64e4i·7-s + (4.09e4 + 8.31e4i)8-s + 7.59e3i·9-s + (−3.69e5 + 5.03e5i)10-s + (4.77e5 + 4.77e5i)11-s + (3.89e5 − 7.47e5i)12-s + (6.65e5 − 6.65e5i)13-s + (−7.35e5 + 1.13e5i)14-s + 5.68e6·15-s + (3.43e6 − 2.40e6i)16-s − 5.80e6·17-s + ⋯
L(s)  = 1  + (−0.151 − 0.988i)2-s + (−0.691 + 0.691i)3-s + (−0.953 + 0.300i)4-s + (−1.39 − 1.39i)5-s + (0.788 + 0.578i)6-s − 0.369i·7-s + (0.441 + 0.897i)8-s + 0.0428i·9-s + (−1.16 + 1.59i)10-s + (0.894 + 0.894i)11-s + (0.452 − 0.867i)12-s + (0.497 − 0.497i)13-s + (−0.365 + 0.0561i)14-s + 1.93·15-s + (0.819 − 0.572i)16-s − 0.991·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.976 - 0.215i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.976 - 0.215i)$
$L(6)$  $\approx$  $0.591086 + 0.0644256i$
$L(\frac12)$  $\approx$  $0.591086 + 0.0644256i$
$L(\frac{13}{2})$   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 \( 1 + (6.87 + 44.7i)T \)
good3 \( 1 + (291. - 291. i)T - 1.77e5iT^{2} \)
5 \( 1 + (9.75e3 + 9.75e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 1.64e4iT - 1.97e9T^{2} \)
11 \( 1 + (-4.77e5 - 4.77e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-6.65e5 + 6.65e5i)T - 1.79e12iT^{2} \)
17 \( 1 + 5.80e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.91e6 + 1.91e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 1.46e7iT - 9.52e14T^{2} \)
29 \( 1 + (-5.83e7 + 5.83e7i)T - 1.22e16iT^{2} \)
31 \( 1 - 8.50e7T + 2.54e16T^{2} \)
37 \( 1 + (-2.52e7 - 2.52e7i)T + 1.77e17iT^{2} \)
41 \( 1 - 9.24e8iT - 5.50e17T^{2} \)
43 \( 1 + (-1.00e9 - 1.00e9i)T + 9.29e17iT^{2} \)
47 \( 1 + 2.86e9T + 2.47e18T^{2} \)
53 \( 1 + (-2.34e9 - 2.34e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (5.55e8 + 5.55e8i)T + 3.01e19iT^{2} \)
61 \( 1 + (2.31e9 - 2.31e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-1.02e10 + 1.02e10i)T - 1.22e20iT^{2} \)
71 \( 1 + 1.39e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.22e10iT - 3.13e20T^{2} \)
79 \( 1 - 3.84e10T + 7.47e20T^{2} \)
83 \( 1 + (4.10e9 - 4.10e9i)T - 1.28e21iT^{2} \)
89 \( 1 - 1.47e10iT - 2.77e21T^{2} \)
97 \( 1 + 9.56e10T + 7.15e21T^{2} \)
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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

−16.61155345688729975558247129858, −15.45849866108879202889339067957, −13.19220433221970275822488914824, −11.96437200982129139274051430067, −11.10987030010882185761883319020, −9.467490812243759060843905066770, −8.051858971059591550111260742865, −4.78799471396137892998727363052, −4.05992341204255417957734698531, −0.995934483925858543306471467300, 0.41382110665762507734977227550, 3.80308871922283397934015659951, 6.30378920264518332829792443796, 7.01062721938748318398556172146, 8.598577710935893186635088828787, 10.98223089738072294385551404440, 12.06393291096726491026599303305, 14.02655667455599218732341706899, 15.17421601112462452383760919539, 16.25052344069052273767318713261

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