Properties

Label 2-2e4-16.11-c6-0-9
Degree $2$
Conductor $16$
Sign $-0.187 + 0.982i$
Analytic cond. $3.68086$
Root an. cond. $1.91855$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.98 − 0.407i)2-s + (−33.4 − 33.4i)3-s + (63.6 − 6.51i)4-s + (−93.3 − 93.3i)5-s + (−280. − 253. i)6-s + 261.·7-s + (506. − 77.9i)8-s + 1.50e3i·9-s + (−783. − 707. i)10-s + (1.37e3 − 1.37e3i)11-s + (−2.34e3 − 1.90e3i)12-s + (−1.49e3 + 1.49e3i)13-s + (2.08e3 − 106. i)14-s + 6.23e3i·15-s + (4.01e3 − 829. i)16-s + 1.84e3·17-s + ⋯
L(s)  = 1  + (0.998 − 0.0509i)2-s + (−1.23 − 1.23i)3-s + (0.994 − 0.101i)4-s + (−0.746 − 0.746i)5-s + (−1.29 − 1.17i)6-s + 0.761·7-s + (0.988 − 0.152i)8-s + 2.06i·9-s + (−0.783 − 0.707i)10-s + (1.03 − 1.03i)11-s + (−1.35 − 1.10i)12-s + (−0.678 + 0.678i)13-s + (0.760 − 0.0387i)14-s + 1.84i·15-s + (0.979 − 0.202i)16-s + 0.375·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(3.68086\)
Root analytic conductor: \(1.91855\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.08449 - 1.31135i\)
\(L(\frac12)\) \(\approx\) \(1.08449 - 1.31135i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.98 + 0.407i)T \)
good3 \( 1 + (33.4 + 33.4i)T + 729iT^{2} \)
5 \( 1 + (93.3 + 93.3i)T + 1.56e4iT^{2} \)
7 \( 1 - 261.T + 1.17e5T^{2} \)
11 \( 1 + (-1.37e3 + 1.37e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (1.49e3 - 1.49e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 1.84e3T + 2.41e7T^{2} \)
19 \( 1 + (-139. - 139. i)T + 4.70e7iT^{2} \)
23 \( 1 - 4.37e3T + 1.48e8T^{2} \)
29 \( 1 + (3.65e3 - 3.65e3i)T - 5.94e8iT^{2} \)
31 \( 1 + 1.83e4iT - 8.87e8T^{2} \)
37 \( 1 + (-4.81e4 - 4.81e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 2.54e4iT - 4.75e9T^{2} \)
43 \( 1 + (8.60e4 - 8.60e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 1.48e5iT - 1.07e10T^{2} \)
53 \( 1 + (-7.99e4 - 7.99e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (-2.02e5 + 2.02e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (6.90e3 - 6.90e3i)T - 5.15e10iT^{2} \)
67 \( 1 + (2.81e5 + 2.81e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 1.38e5T + 1.28e11T^{2} \)
73 \( 1 - 3.81e5iT - 1.51e11T^{2} \)
79 \( 1 + 3.59e5iT - 2.43e11T^{2} \)
83 \( 1 + (1.95e5 + 1.95e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.10e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.65e5T + 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.06542815641123236627809003975, −16.39693667325039386636184337646, −14.39880228128784473241324128206, −12.99333874171326787462713299031, −11.78332431124346481056383474409, −11.41550598882033340769917791019, −7.83593632503456052565388039032, −6.31890607838481437001110677338, −4.74598429258803943405460784656, −1.19826278157570627641269874128, 3.92276577852914341595327274642, 5.22013145983549651161515164178, 7.06822668478478936554141408887, 10.22295109759800744659224619473, 11.36896278183733800530432421266, 12.14915299152943283588299386726, 14.79779378426987042947964651269, 15.14160132426790788612651659086, 16.59745126623085876074208199599, 17.66666156381241656445602895646

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