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Properties

Label 2-2e4-16.3-c10-0-16
Degree $2$
Conductor $16$
Sign $-0.253 + 0.967i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−25.9 + 18.6i)2-s + (280. − 280. i)3-s + (326. − 970. i)4-s + (3.10e3 − 3.10e3i)5-s + (−2.05e3 + 1.25e4i)6-s − 2.74e4·7-s + (9.63e3 + 3.13e4i)8-s − 9.83e4i·9-s + (−2.26e4 + 1.38e5i)10-s + (5.63e4 + 5.63e4i)11-s + (−1.80e5 − 3.63e5i)12-s + (−1.81e5 − 1.81e5i)13-s + (7.12e5 − 5.11e5i)14-s − 1.74e6i·15-s + (−8.35e5 − 6.33e5i)16-s − 1.80e5·17-s + ⋯
L(s)  = 1  + (−0.812 + 0.583i)2-s + (1.15 − 1.15i)3-s + (0.318 − 0.947i)4-s + (0.992 − 0.992i)5-s + (−0.263 + 1.61i)6-s − 1.63·7-s + (0.294 + 0.955i)8-s − 1.66i·9-s + (−0.226 + 1.38i)10-s + (0.349 + 0.349i)11-s + (−0.725 − 1.46i)12-s + (−0.489 − 0.489i)13-s + (1.32 − 0.951i)14-s − 2.29i·15-s + (−0.796 − 0.604i)16-s − 0.127·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.253 + 0.967i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.253 + 0.967i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.940902 - 1.21954i\)
\(L(\frac12)\) \(\approx\) \(0.940902 - 1.21954i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (25.9 - 18.6i)T \)
good3 \( 1 + (-280. + 280. i)T - 5.90e4iT^{2} \)
5 \( 1 + (-3.10e3 + 3.10e3i)T - 9.76e6iT^{2} \)
7 \( 1 + 2.74e4T + 2.82e8T^{2} \)
11 \( 1 + (-5.63e4 - 5.63e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (1.81e5 + 1.81e5i)T + 1.37e11iT^{2} \)
17 \( 1 + 1.80e5T + 2.01e12T^{2} \)
19 \( 1 + (1.21e5 - 1.21e5i)T - 6.13e12iT^{2} \)
23 \( 1 - 7.96e6T + 4.14e13T^{2} \)
29 \( 1 + (2.05e7 + 2.05e7i)T + 4.20e14iT^{2} \)
31 \( 1 + 1.51e7iT - 8.19e14T^{2} \)
37 \( 1 + (-5.65e7 + 5.65e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 2.70e7iT - 1.34e16T^{2} \)
43 \( 1 + (-2.01e8 - 2.01e8i)T + 2.16e16iT^{2} \)
47 \( 1 + 1.85e8iT - 5.25e16T^{2} \)
53 \( 1 + (1.90e8 - 1.90e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (-5.85e8 - 5.85e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-4.76e8 - 4.76e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (1.01e7 - 1.01e7i)T - 1.82e18iT^{2} \)
71 \( 1 + 3.69e8T + 3.25e18T^{2} \)
73 \( 1 + 2.59e9iT - 4.29e18T^{2} \)
79 \( 1 - 2.32e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.58e9 + 3.58e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 1.93e9iT - 3.11e19T^{2} \)
97 \( 1 - 5.12e9T + 7.37e19T^{2} \)
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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

−16.54811031476166468399193254282, −14.93054090085785217241827711074, −13.43564648195199827457677215411, −12.74905209361091752089055945694, −9.647148673411246817091799991790, −9.031144424319230561669497305862, −7.39219649463917950947388820348, −6.05309564584598351661763281704, −2.38167387782610028085711619505, −0.813332304120453270419277358574, 2.57081308750754808232382322567, 3.47857573432921232720495368349, 6.81204757191563027043985461367, 9.109781550705999634979013227343, 9.702679840607813235621532139856, 10.73117623490083316287925876638, 13.10456178383070128577103919039, 14.42760142941664661568863954589, 15.85999130418453106720793604416, 16.98161681424891322987638015269

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