↑

Properties

Label 2-2e4-16.11-c12-0-2
Degree $2$
Conductor $16$
Sign $-0.766 + 0.642i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−50.3 + 39.5i)2-s + (739. + 739. i)3-s + (973. − 3.97e3i)4-s + (−805. − 805. i)5-s + (−6.64e4 − 8.01e3i)6-s − 9.75e4·7-s + (1.08e5 + 2.38e5i)8-s + 5.61e5i·9-s + (7.23e4 + 8.72e3i)10-s + (−1.24e6 + 1.24e6i)11-s + (3.66e6 − 2.22e6i)12-s + (−1.01e6 + 1.01e6i)13-s + (4.91e6 − 3.85e6i)14-s − 1.19e6i·15-s + (−1.48e7 − 7.74e6i)16-s − 3.87e7·17-s + ⋯
L(s)  = 1  + (−0.786 + 0.617i)2-s + (1.01 + 1.01i)3-s + (0.237 − 0.971i)4-s + (−0.0515 − 0.0515i)5-s + (−1.42 − 0.171i)6-s − 0.829·7-s + (0.412 + 0.910i)8-s + 1.05i·9-s + (0.0723 + 0.00872i)10-s + (−0.703 + 0.703i)11-s + (1.22 − 0.743i)12-s + (−0.211 + 0.211i)13-s + (0.652 − 0.512i)14-s − 0.104i·15-s + (−0.886 − 0.461i)16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.183177 - 0.503323i\)
\(L(\frac12)\) \(\approx\) \(0.183177 - 0.503323i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (50.3 - 39.5i)T \)
good3 \( 1 + (-739. - 739. i)T + 5.31e5iT^{2} \)
5 \( 1 + (805. + 805. i)T + 2.44e8iT^{2} \)
7 \( 1 + 9.75e4T + 1.38e10T^{2} \)
11 \( 1 + (1.24e6 - 1.24e6i)T - 3.13e12iT^{2} \)
13 \( 1 + (1.01e6 - 1.01e6i)T - 2.32e13iT^{2} \)
17 \( 1 + 3.87e7T + 5.82e14T^{2} \)
19 \( 1 + (2.64e7 + 2.64e7i)T + 2.21e15iT^{2} \)
23 \( 1 - 1.86e8T + 2.19e16T^{2} \)
29 \( 1 + (5.99e8 - 5.99e8i)T - 3.53e17iT^{2} \)
31 \( 1 + 1.96e7iT - 7.87e17T^{2} \)
37 \( 1 + (-7.61e8 - 7.61e8i)T + 6.58e18iT^{2} \)
41 \( 1 + 3.33e8iT - 2.25e19T^{2} \)
43 \( 1 + (7.52e9 - 7.52e9i)T - 3.99e19iT^{2} \)
47 \( 1 + 1.17e10iT - 1.16e20T^{2} \)
53 \( 1 + (-2.62e10 - 2.62e10i)T + 4.91e20iT^{2} \)
59 \( 1 + (3.50e10 - 3.50e10i)T - 1.77e21iT^{2} \)
61 \( 1 + (3.55e10 - 3.55e10i)T - 2.65e21iT^{2} \)
67 \( 1 + (-3.63e10 - 3.63e10i)T + 8.18e21iT^{2} \)
71 \( 1 - 1.95e11T + 1.64e22T^{2} \)
73 \( 1 - 1.16e11iT - 2.29e22T^{2} \)
79 \( 1 + 6.91e10iT - 5.90e22T^{2} \)
83 \( 1 + (2.73e11 + 2.73e11i)T + 1.06e23iT^{2} \)
89 \( 1 - 7.49e11iT - 2.46e23T^{2} \)
97 \( 1 - 1.33e11T + 6.93e23T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.66175417514447268308274936588, −15.52922461401785604539052537075, −14.92290187982603851799864926919, −13.28574511899852400814350248878, −10.70311814830911694470483831896, −9.542122176799949292683825356889, −8.641670341075923264088679619027, −6.88311907448753391968209587521, −4.64529517683647130007490274374, −2.53926332704365598087320655268, 0.22845394430505371868972042140, 2.07390943679706075247643228109, 3.27581497588061031459009960077, 6.84329977179762524706073322441, 8.134675249605968445976232186579, 9.276753244416017512708602698053, 11.01388823592703247205074692564, 12.85483530758363067845439888903, 13.37599854093344943179200642181, 15.40161559946076394554866522561

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