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Properties

Label 2-2e4-16.3-c4-0-0
Degree $2$
Conductor $16$
Sign $-0.573 - 0.819i$
Analytic cond. $1.65391$
Root an. cond. $1.28604$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.97 − 2.67i)2-s + (−9.42 + 9.42i)3-s + (1.73 + 15.9i)4-s + (−2.84 + 2.84i)5-s + (53.2 − 2.90i)6-s − 76.7·7-s + (37.2 − 52.0i)8-s − 96.6i·9-s + (16.0 − 0.876i)10-s + (121. + 121. i)11-s + (−166. − 133. i)12-s + (27.1 + 27.1i)13-s + (228. + 205. i)14-s − 53.6i·15-s + (−249. + 55.2i)16-s − 88.0·17-s + ⋯
L(s)  = 1  + (−0.744 − 0.667i)2-s + (−1.04 + 1.04i)3-s + (0.108 + 0.994i)4-s + (−0.113 + 0.113i)5-s + (1.47 − 0.0805i)6-s − 1.56·7-s + (0.582 − 0.812i)8-s − 1.19i·9-s + (0.160 − 0.00876i)10-s + (1.00 + 1.00i)11-s + (−1.15 − 0.927i)12-s + (0.160 + 0.160i)13-s + (1.16 + 1.04i)14-s − 0.238i·15-s + (−0.976 + 0.215i)16-s − 0.304·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(1.65391\)
Root analytic conductor: \(1.28604\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ -0.573 - 0.819i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.153023 + 0.293788i\)
\(L(\frac12)\) \(\approx\) \(0.153023 + 0.293788i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.97 + 2.67i)T \)
good3 \( 1 + (9.42 - 9.42i)T - 81iT^{2} \)
5 \( 1 + (2.84 - 2.84i)T - 625iT^{2} \)
7 \( 1 + 76.7T + 2.40e3T^{2} \)
11 \( 1 + (-121. - 121. i)T + 1.46e4iT^{2} \)
13 \( 1 + (-27.1 - 27.1i)T + 2.85e4iT^{2} \)
17 \( 1 + 88.0T + 8.35e4T^{2} \)
19 \( 1 + (261. - 261. i)T - 1.30e5iT^{2} \)
23 \( 1 - 93.4T + 2.79e5T^{2} \)
29 \( 1 + (-272. - 272. i)T + 7.07e5iT^{2} \)
31 \( 1 + 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + (1.04e3 - 1.04e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 915. iT - 2.82e6T^{2} \)
43 \( 1 + (-1.11e3 - 1.11e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 1.72e3iT - 4.87e6T^{2} \)
53 \( 1 + (-734. + 734. i)T - 7.89e6iT^{2} \)
59 \( 1 + (1.20e3 + 1.20e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-580. - 580. i)T + 1.38e7iT^{2} \)
67 \( 1 + (1.48e3 - 1.48e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 5.57e3T + 2.54e7T^{2} \)
73 \( 1 - 6.61e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.55e3 - 2.55e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.71e3T + 8.85e7T^{2} \)
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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

−18.88215812172987316227102017107, −17.24322319949097097911966314607, −16.57883045829110248387340368983, −15.41902512507524921324010746347, −12.79424773959630429985875734893, −11.58000329491176468100633440061, −10.19718361834186652619799523679, −9.370826893116514104592579016593, −6.61361030650826395055363265248, −3.93824887074054546169436152035, 0.45980600418400553558267867558, 6.08346998053478621415460806589, 6.84670409430693758411867543178, 8.930718607438399315631159596087, 10.78409429424232340969181123381, 12.33045747510064417017908147345, 13.74306542803189267521688106082, 15.85301342514478290606478918765, 16.77131495106437194003516168885, 17.77270223990790180568924083654

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