Properties

Label 2-2e4-16.3-c12-0-21
Degree $2$
Conductor $16$
Sign $-0.921 + 0.389i$
Analytic cond. $14.6239$
Root an. cond. $3.82412$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (24.3 − 59.1i)2-s + (949. − 949. i)3-s + (−2.90e3 − 2.88e3i)4-s + (1.07e4 − 1.07e4i)5-s + (−3.30e4 − 7.93e4i)6-s + 3.58e4·7-s + (−2.41e5 + 1.01e5i)8-s − 1.27e6i·9-s + (−3.74e5 − 8.98e5i)10-s + (2.14e6 + 2.14e6i)11-s + (−5.49e6 + 1.99e4i)12-s + (4.27e6 + 4.27e6i)13-s + (8.73e5 − 2.11e6i)14-s − 2.04e7i·15-s + (1.21e5 + 1.67e7i)16-s − 1.30e7·17-s + ⋯
L(s)  = 1  + (0.381 − 0.924i)2-s + (1.30 − 1.30i)3-s + (−0.709 − 0.704i)4-s + (0.688 − 0.688i)5-s + (−0.707 − 1.70i)6-s + 0.304·7-s + (−0.921 + 0.387i)8-s − 2.39i·9-s + (−0.374 − 0.898i)10-s + (1.21 + 1.21i)11-s + (−1.84 + 0.00669i)12-s + (0.885 + 0.885i)13-s + (0.116 − 0.281i)14-s − 1.79i·15-s + (0.00726 + 0.999i)16-s − 0.539·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.921 + 0.389i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ -0.921 + 0.389i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.721289 - 3.55856i\)
\(L(\frac12)\) \(\approx\) \(0.721289 - 3.55856i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-24.3 + 59.1i)T \)
good3 \( 1 + (-949. + 949. i)T - 5.31e5iT^{2} \)
5 \( 1 + (-1.07e4 + 1.07e4i)T - 2.44e8iT^{2} \)
7 \( 1 - 3.58e4T + 1.38e10T^{2} \)
11 \( 1 + (-2.14e6 - 2.14e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (-4.27e6 - 4.27e6i)T + 2.32e13iT^{2} \)
17 \( 1 + 1.30e7T + 5.82e14T^{2} \)
19 \( 1 + (3.07e7 - 3.07e7i)T - 2.21e15iT^{2} \)
23 \( 1 + 1.14e8T + 2.19e16T^{2} \)
29 \( 1 + (-2.27e8 - 2.27e8i)T + 3.53e17iT^{2} \)
31 \( 1 + 1.49e8iT - 7.87e17T^{2} \)
37 \( 1 + (1.72e8 - 1.72e8i)T - 6.58e18iT^{2} \)
41 \( 1 + 4.72e8iT - 2.25e19T^{2} \)
43 \( 1 + (-4.82e9 - 4.82e9i)T + 3.99e19iT^{2} \)
47 \( 1 + 2.13e10iT - 1.16e20T^{2} \)
53 \( 1 + (-6.58e9 + 6.58e9i)T - 4.91e20iT^{2} \)
59 \( 1 + (-9.14e9 - 9.14e9i)T + 1.77e21iT^{2} \)
61 \( 1 + (5.90e10 + 5.90e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (1.06e10 - 1.06e10i)T - 8.18e21iT^{2} \)
71 \( 1 + 4.22e10T + 1.64e22T^{2} \)
73 \( 1 - 2.02e11iT - 2.29e22T^{2} \)
79 \( 1 + 2.08e10iT - 5.90e22T^{2} \)
83 \( 1 + (-6.85e10 + 6.85e10i)T - 1.06e23iT^{2} \)
89 \( 1 + 3.94e11iT - 2.46e23T^{2} \)
97 \( 1 + 3.92e11T + 6.93e23T^{2} \)
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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

−14.74380063838883917890107390669, −13.90140328138246559678007725655, −12.92265179650657337062349210190, −11.93157683727338635990286777039, −9.493789586421351251330038895639, −8.600126576082882191113164034299, −6.49824853683248138730107496520, −4.02917530104289218942632040679, −1.95402056327839909369290964303, −1.43615008324449785099605406211, 2.94973087898131273093841759269, 4.17888274670165529014049082057, 6.09572412425905659308444505849, 8.264475001261527401391768729432, 9.205487565196749462646623738009, 10.76512300691671021503122292275, 13.61526886772106023266313207241, 14.21284622701147587325126520311, 15.23662409319070451017215463399, 16.25891322832450341690450482356

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