Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.998 + 0.0551i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 + 0.316i)2-s + (−3.27 − 3.27i)3-s + (7.80 + 1.77i)4-s + (−12.6 + 12.6i)5-s + (−8.16 − 10.2i)6-s − 13.8i·7-s + (21.3 + 7.45i)8-s − 5.59i·9-s + (−39.5 + 31.5i)10-s + (1.54 − 1.54i)11-s + (−19.7 − 31.3i)12-s + (32.7 + 32.7i)13-s + (4.38 − 38.9i)14-s + 82.7·15-s + (57.6 + 27.7i)16-s + 18.6·17-s + ⋯
L(s)  = 1  + (0.993 + 0.111i)2-s + (−0.629 − 0.629i)3-s + (0.975 + 0.222i)4-s + (−1.13 + 1.13i)5-s + (−0.555 − 0.695i)6-s − 0.749i·7-s + (0.944 + 0.329i)8-s − 0.207i·9-s + (−1.25 + 0.997i)10-s + (0.0424 − 0.0424i)11-s + (−0.474 − 0.753i)12-s + (0.699 + 0.699i)13-s + (0.0837 − 0.744i)14-s + 1.42·15-s + (0.901 + 0.433i)16-s + 0.266·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0551i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0551i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.998 + 0.0551i$
motivic weight  =  \(3\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :3/2),\ 0.998 + 0.0551i)$
$L(2)$  $\approx$  $1.29158 - 0.0356701i$
$L(\frac12)$  $\approx$  $1.29158 - 0.0356701i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-2.81 - 0.316i)T \)
good3 \( 1 + (3.27 + 3.27i)T + 27iT^{2} \)
5 \( 1 + (12.6 - 12.6i)T - 125iT^{2} \)
7 \( 1 + 13.8iT - 343T^{2} \)
11 \( 1 + (-1.54 + 1.54i)T - 1.33e3iT^{2} \)
13 \( 1 + (-32.7 - 32.7i)T + 2.19e3iT^{2} \)
17 \( 1 - 18.6T + 4.91e3T^{2} \)
19 \( 1 + (86.4 + 86.4i)T + 6.85e3iT^{2} \)
23 \( 1 - 134. iT - 1.21e4T^{2} \)
29 \( 1 + (59.7 + 59.7i)T + 2.43e4iT^{2} \)
31 \( 1 + 31.5T + 2.97e4T^{2} \)
37 \( 1 + (-89.1 + 89.1i)T - 5.06e4iT^{2} \)
41 \( 1 - 210. iT - 6.89e4T^{2} \)
43 \( 1 + (-119. + 119. i)T - 7.95e4iT^{2} \)
47 \( 1 + 182.T + 1.03e5T^{2} \)
53 \( 1 + (26.1 - 26.1i)T - 1.48e5iT^{2} \)
59 \( 1 + (-441. + 441. i)T - 2.05e5iT^{2} \)
61 \( 1 + (174. + 174. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-91.7 - 91.7i)T + 3.00e5iT^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 - 299. iT - 3.89e5T^{2} \)
79 \( 1 + 943.T + 4.93e5T^{2} \)
83 \( 1 + (-313. - 313. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−18.80428636821950363259418671615, −17.24409571198555303481118615532, −15.75319592822826627784935850727, −14.62429414166078430699001073400, −13.20205984725613540839392196906, −11.65697398613831378058459120435, −11.00004024031805071090457338786, −7.43192695854085614181681259807, −6.47822961820532445296571330633, −3.81746618896785242225556130638, 4.23731031807583349022840487675, 5.62036500732862361628891850858, 8.248769012487150467291391348964, 10.67481878802626459797891779872, 11.96202222694314605717937679653, 12.87078476741698750657766775968, 14.96433181711139586642192676086, 16.01113049375911288393669519023, 16.67359888057166418265649385568, 19.01838602946845692153318283135

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