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Properties

Label 2-2e4-16.13-c13-0-20
Degree $2$
Conductor $16$
Sign $0.989 + 0.142i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (54.8 + 71.9i)2-s + (242. + 242. i)3-s + (−2.16e3 + 7.89e3i)4-s + (4.24e4 − 4.24e4i)5-s + (−4.14e3 + 3.07e4i)6-s − 4.21e5i·7-s + (−6.87e5 + 2.77e5i)8-s − 1.47e6i·9-s + (5.38e6 + 7.25e5i)10-s + (2.37e6 − 2.37e6i)11-s + (−2.43e6 + 1.38e6i)12-s + (−2.71e6 − 2.71e6i)13-s + (3.03e7 − 2.31e7i)14-s + 2.05e7·15-s + (−5.77e7 − 3.42e7i)16-s − 4.61e7·17-s + ⋯
L(s)  = 1  + (0.606 + 0.795i)2-s + (0.191 + 0.191i)3-s + (−0.264 + 0.964i)4-s + (1.21 − 1.21i)5-s + (−0.0362 + 0.268i)6-s − 1.35i·7-s + (−0.927 + 0.374i)8-s − 0.926i·9-s + (1.70 + 0.229i)10-s + (0.403 − 0.403i)11-s + (−0.235 + 0.134i)12-s + (−0.156 − 0.156i)13-s + (1.07 − 0.821i)14-s + 0.465·15-s + (−0.859 − 0.510i)16-s − 0.463·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ 0.989 + 0.142i)\)

Particular Values

\(L(7)\) \(\approx\) \(3.07233 - 0.220035i\)
\(L(\frac12)\) \(\approx\) \(3.07233 - 0.220035i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-54.8 - 71.9i)T \)
good3 \( 1 + (-242. - 242. i)T + 1.59e6iT^{2} \)
5 \( 1 + (-4.24e4 + 4.24e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 4.21e5iT - 9.68e10T^{2} \)
11 \( 1 + (-2.37e6 + 2.37e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (2.71e6 + 2.71e6i)T + 3.02e14iT^{2} \)
17 \( 1 + 4.61e7T + 9.90e15T^{2} \)
19 \( 1 + (-1.84e8 - 1.84e8i)T + 4.20e16iT^{2} \)
23 \( 1 - 3.33e8iT - 5.04e17T^{2} \)
29 \( 1 + (-3.25e9 - 3.25e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 5.55e8T + 2.44e19T^{2} \)
37 \( 1 + (1.90e10 - 1.90e10i)T - 2.43e20iT^{2} \)
41 \( 1 + 5.18e10iT - 9.25e20T^{2} \)
43 \( 1 + (3.91e10 - 3.91e10i)T - 1.71e21iT^{2} \)
47 \( 1 - 4.84e10T + 5.46e21T^{2} \)
53 \( 1 + (-1.98e10 + 1.98e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (-3.57e11 + 3.57e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (-1.35e10 - 1.35e10i)T + 1.61e23iT^{2} \)
67 \( 1 + (-7.55e11 - 7.55e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 7.59e11iT - 1.16e24T^{2} \)
73 \( 1 - 1.37e10iT - 1.67e24T^{2} \)
79 \( 1 - 2.36e12T + 4.66e24T^{2} \)
83 \( 1 + (-2.54e12 - 2.54e12i)T + 8.87e24iT^{2} \)
89 \( 1 - 1.69e12iT - 2.19e25T^{2} \)
97 \( 1 - 2.33e12T + 6.73e25T^{2} \)
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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

−16.02536669333533786947166138365, −14.22221435349660010276800233412, −13.52958400532453979492210743728, −12.25761510720884187045819183325, −9.836791441421837686675751732186, −8.575257691768416301661817639053, −6.68818184498136185644154517883, −5.20057433174192910855139144051, −3.71045649795186914394141341588, −1.01450761263047804786402542398, 2.03572112977210855091591767931, 2.70009987580076073657513364008, 5.22446819866109150349717736437, 6.56543110261496848264019249079, 9.207599950157964340502071731055, 10.44073252096199204241345477259, 11.78971602519689313426214103924, 13.36908583271800574981237081288, 14.27283427980811797948304913961, 15.39590930727290811028217535276

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