Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.515 - 0.856i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.29 + 1.65i)2-s + (−5.96 + 5.96i)3-s + (2.50 − 7.59i)4-s + (8.67 + 8.67i)5-s + (3.77 − 23.5i)6-s + 1.63i·7-s + (6.86 + 21.5i)8-s − 44.1i·9-s + (−34.2 − 5.49i)10-s + (18.2 + 18.2i)11-s + (30.4 + 60.2i)12-s + (−9.34 + 9.34i)13-s + (−2.71 − 3.75i)14-s − 103.·15-s + (−51.4 − 38.0i)16-s + 53.6·17-s + ⋯
L(s)  = 1  + (−0.810 + 0.586i)2-s + (−1.14 + 1.14i)3-s + (0.312 − 0.949i)4-s + (0.776 + 0.776i)5-s + (0.257 − 1.60i)6-s + 0.0885i·7-s + (0.303 + 0.952i)8-s − 1.63i·9-s + (−1.08 − 0.173i)10-s + (0.498 + 0.498i)11-s + (0.731 + 1.44i)12-s + (−0.199 + 0.199i)13-s + (−0.0518 − 0.0717i)14-s − 1.78·15-s + (−0.804 − 0.594i)16-s + 0.764·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.515 - 0.856i$
motivic weight  =  \(3\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :3/2),\ -0.515 - 0.856i)$
$L(2)$  $\approx$  $0.282044 + 0.499003i$
$L(\frac12)$  $\approx$  $0.282044 + 0.499003i$
$L(\frac{5}{2})$   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 \( 1 + (2.29 - 1.65i)T \)
good3 \( 1 + (5.96 - 5.96i)T - 27iT^{2} \)
5 \( 1 + (-8.67 - 8.67i)T + 125iT^{2} \)
7 \( 1 - 1.63iT - 343T^{2} \)
11 \( 1 + (-18.2 - 18.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (9.34 - 9.34i)T - 2.19e3iT^{2} \)
17 \( 1 - 53.6T + 4.91e3T^{2} \)
19 \( 1 + (-70.9 + 70.9i)T - 6.85e3iT^{2} \)
23 \( 1 + 25.1iT - 1.21e4T^{2} \)
29 \( 1 + (181. - 181. i)T - 2.43e4iT^{2} \)
31 \( 1 - 132.T + 2.97e4T^{2} \)
37 \( 1 + (-174. - 174. i)T + 5.06e4iT^{2} \)
41 \( 1 + 198. iT - 6.89e4T^{2} \)
43 \( 1 + (285. + 285. i)T + 7.95e4iT^{2} \)
47 \( 1 - 78.3T + 1.03e5T^{2} \)
53 \( 1 + (525. + 525. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-46.5 - 46.5i)T + 2.05e5iT^{2} \)
61 \( 1 + (-193. + 193. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-282. + 282. i)T - 3.00e5iT^{2} \)
71 \( 1 - 727. iT - 3.57e5T^{2} \)
73 \( 1 + 106. iT - 3.89e5T^{2} \)
79 \( 1 + 58.9T + 4.93e5T^{2} \)
83 \( 1 + (410. - 410. i)T - 5.71e5iT^{2} \)
89 \( 1 + 768. iT - 7.04e5T^{2} \)
97 \( 1 + 809.T + 9.12e5T^{2} \)
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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

−18.55591025025149052524263731487, −17.52594130393967864454040002522, −16.71171042161424593286110844091, −15.47385542628266901040824855860, −14.34517882687107856052799216409, −11.55298207393696286132009853932, −10.32771548353112740908687031961, −9.477681158678570042125916994194, −6.74697335752028892035439759222, −5.31112394515550259827808581077, 1.22196100417709754142613348311, 5.91305368120038529267052862920, 7.74657942407784517463432430670, 9.675786585896418739096110158090, 11.40897587840056563710395050447, 12.42647682748603329334970290339, 13.49315262608994140882136299642, 16.56015012857676819898551060375, 17.12161523152205369607742367113, 18.17303671270217465371053883425

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