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Properties

Label 2-2e4-16.3-c10-0-13
Degree $2$
Conductor $16$
Sign $-0.510 + 0.859i$
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  + (−16.8 − 27.2i)2-s + (183. − 183. i)3-s + (−458. + 915. i)4-s + (3.49e3 − 3.49e3i)5-s + (−8.09e3 − 1.91e3i)6-s + 2.61e4·7-s + (3.26e4 − 2.90e3i)8-s − 8.51e3i·9-s + (−1.53e5 − 3.64e4i)10-s + (−2.36e4 − 2.36e4i)11-s + (8.39e4 + 2.52e5i)12-s + (−1.98e5 − 1.98e5i)13-s + (−4.39e5 − 7.11e5i)14-s − 1.28e6i·15-s + (−6.27e5 − 8.39e5i)16-s − 3.71e4·17-s + ⋯
L(s)  = 1  + (−0.525 − 0.850i)2-s + (0.756 − 0.756i)3-s + (−0.447 + 0.894i)4-s + (1.11 − 1.11i)5-s + (−1.04 − 0.246i)6-s + 1.55·7-s + (0.996 − 0.0885i)8-s − 0.144i·9-s + (−1.53 − 0.364i)10-s + (−0.146 − 0.146i)11-s + (0.337 + 1.01i)12-s + (−0.534 − 0.534i)13-s + (−0.816 − 1.32i)14-s − 1.69i·15-s + (−0.598 − 0.801i)16-s − 0.0261·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.510 + 0.859i$
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: \((2,\ 16,\ (\ :5),\ -0.510 + 0.859i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.05761 - 1.85907i\)
\(L(\frac12)\) \(\approx\) \(1.05761 - 1.85907i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (16.8 + 27.2i)T \)
good3 \( 1 + (-183. + 183. i)T - 5.90e4iT^{2} \)
5 \( 1 + (-3.49e3 + 3.49e3i)T - 9.76e6iT^{2} \)
7 \( 1 - 2.61e4T + 2.82e8T^{2} \)
11 \( 1 + (2.36e4 + 2.36e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (1.98e5 + 1.98e5i)T + 1.37e11iT^{2} \)
17 \( 1 + 3.71e4T + 2.01e12T^{2} \)
19 \( 1 + (2.46e6 - 2.46e6i)T - 6.13e12iT^{2} \)
23 \( 1 + 9.28e5T + 4.14e13T^{2} \)
29 \( 1 + (-2.31e7 - 2.31e7i)T + 4.20e14iT^{2} \)
31 \( 1 + 4.20e7iT - 8.19e14T^{2} \)
37 \( 1 + (7.15e7 - 7.15e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 3.19e7iT - 1.34e16T^{2} \)
43 \( 1 + (7.38e7 + 7.38e7i)T + 2.16e16iT^{2} \)
47 \( 1 + 4.97e6iT - 5.25e16T^{2} \)
53 \( 1 + (4.22e8 - 4.22e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (2.04e8 + 2.04e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-9.65e8 - 9.65e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (4.22e7 - 4.22e7i)T - 1.82e18iT^{2} \)
71 \( 1 - 1.24e9T + 3.25e18T^{2} \)
73 \( 1 - 1.86e8iT - 4.29e18T^{2} \)
79 \( 1 + 3.24e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.80e9 + 1.80e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 1.74e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.21e10T + 7.37e19T^{2} \)
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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

−16.89979533019532162893791072128, −14.31210448343476836982633654197, −13.30201023695092188641268577675, −12.24120614143066184466532363909, −10.39578277400319738706244242543, −8.717820304632127044276220298478, −7.964150610018473970286842349694, −4.93561849002348737962882239025, −2.16671548343905612429176626793, −1.30484427699679663775312332397, 2.12226380487328081279164261994, 4.79820858569671817296510684776, 6.71746612117539948432229029962, 8.457643737740486601447334389631, 9.772888344551055759597523723253, 10.86050636871714132372264186520, 14.04541948311443095062497209072, 14.48901924085005822278099086383, 15.46003757666786692346638473728, 17.44053742212137538152538124209

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