↑

Properties

Label 2-2e4-16.11-c10-0-11
Degree $2$
Conductor $16$
Sign $0.925 + 0.378i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (12.2 + 29.5i)2-s + (−126. − 126. i)3-s + (−722. + 725. i)4-s + (−1.65e3 − 1.65e3i)5-s + (2.17e3 − 5.27e3i)6-s + 2.98e4·7-s + (−3.03e4 − 1.24e4i)8-s − 2.72e4i·9-s + (2.85e4 − 6.92e4i)10-s + (8.99e4 − 8.99e4i)11-s + (1.82e5 − 391. i)12-s + (5.66e3 − 5.66e3i)13-s + (3.66e5 + 8.82e5i)14-s + 4.17e5i·15-s + (−4.49e3 − 1.04e6i)16-s + 2.33e6·17-s + ⋯
L(s)  = 1  + (0.383 + 0.923i)2-s + (−0.518 − 0.518i)3-s + (−0.705 + 0.708i)4-s + (−0.529 − 0.529i)5-s + (0.280 − 0.678i)6-s + 1.77·7-s + (−0.925 − 0.379i)8-s − 0.461i·9-s + (0.285 − 0.692i)10-s + (0.558 − 0.558i)11-s + (0.733 − 0.00157i)12-s + (0.0152 − 0.0152i)13-s + (0.681 + 1.64i)14-s + 0.549i·15-s + (−0.00428 − 0.999i)16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ 0.925 + 0.378i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.59830 - 0.314361i\)
\(L(\frac12)\) \(\approx\) \(1.59830 - 0.314361i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-12.2 - 29.5i)T \)
good3 \( 1 + (126. + 126. i)T + 5.90e4iT^{2} \)
5 \( 1 + (1.65e3 + 1.65e3i)T + 9.76e6iT^{2} \)
7 \( 1 - 2.98e4T + 2.82e8T^{2} \)
11 \( 1 + (-8.99e4 + 8.99e4i)T - 2.59e10iT^{2} \)
13 \( 1 + (-5.66e3 + 5.66e3i)T - 1.37e11iT^{2} \)
17 \( 1 - 2.33e6T + 2.01e12T^{2} \)
19 \( 1 + (2.56e6 + 2.56e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 2.91e6T + 4.14e13T^{2} \)
29 \( 1 + (4.43e6 - 4.43e6i)T - 4.20e14iT^{2} \)
31 \( 1 + 2.94e7iT - 8.19e14T^{2} \)
37 \( 1 + (-1.06e7 - 1.06e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 5.47e7iT - 1.34e16T^{2} \)
43 \( 1 + (-7.25e7 + 7.25e7i)T - 2.16e16iT^{2} \)
47 \( 1 - 2.46e8iT - 5.25e16T^{2} \)
53 \( 1 + (-3.94e8 - 3.94e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (7.21e8 - 7.21e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (-5.12e8 + 5.12e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-1.09e8 - 1.09e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 1.01e7T + 3.25e18T^{2} \)
73 \( 1 - 2.60e9iT - 4.29e18T^{2} \)
79 \( 1 + 2.96e9iT - 9.46e18T^{2} \)
83 \( 1 + (3.36e9 + 3.36e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 2.18e8iT - 3.11e19T^{2} \)
97 \( 1 + 1.07e10T + 7.37e19T^{2} \)
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

−16.77565298471465540084489217964, −15.14721198277564454139624920932, −14.13020747353196360169098526106, −12.42400862871910608625357905248, −11.50396064342120688338941009530, −8.690236769516716376366818958602, −7.55939872788550170365207408561, −5.78309879958459949354061416544, −4.28620947640305513464650068745, −0.815337656967721325875680969512, 1.67794473160354989458554801302, 4.04409625012060438103233046402, 5.32491600324809959614807033077, 8.038657744551142711967789038187, 10.21213782421414154488979286009, 11.20268089314098821907250947123, 12.12432708166634984202174279813, 14.26154654111142297067479123723, 14.94764217932204156470633033459, 16.94202869730406230534926121321

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