↑

Properties

Label 2-2016-24.5-c2-0-37
Degree $2$
Conductor $2016$
Sign $0.598 + 0.800i$
Analytic cond. $54.9320$
Root an. cond. $7.41161$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.92·5-s + 2.64·7-s + 14.3·11-s + 0.111i·13-s − 7.19i·17-s − 22.5i·19-s − 12.7i·23-s − 21.2·25-s − 6.94·29-s + 49.7·31-s + 5.09·35-s + 16.9i·37-s − 45.1i·41-s − 66.4i·43-s + 14.0i·47-s + ⋯
L(s)  = 1  + 0.385·5-s + 0.377·7-s + 1.30·11-s + 0.00858i·13-s − 0.423i·17-s − 1.18i·19-s − 0.553i·23-s − 0.851·25-s − 0.239·29-s + 1.60·31-s + 0.145·35-s + 0.458i·37-s − 1.10i·41-s − 1.54i·43-s + 0.298i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.598 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.598 + 0.800i$
Analytic conductor: \(54.9320\)
Root analytic conductor: \(7.41161\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1),\ 0.598 + 0.800i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.414091895\)
\(L(\frac12)\) \(\approx\) \(2.414091895\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 - 1.92T + 25T^{2} \)
11 \( 1 - 14.3T + 121T^{2} \)
13 \( 1 - 0.111iT - 169T^{2} \)
17 \( 1 + 7.19iT - 289T^{2} \)
19 \( 1 + 22.5iT - 361T^{2} \)
23 \( 1 + 12.7iT - 529T^{2} \)
29 \( 1 + 6.94T + 841T^{2} \)
31 \( 1 - 49.7T + 961T^{2} \)
37 \( 1 - 16.9iT - 1.36e3T^{2} \)
41 \( 1 + 45.1iT - 1.68e3T^{2} \)
43 \( 1 + 66.4iT - 1.84e3T^{2} \)
47 \( 1 - 14.0iT - 2.20e3T^{2} \)
53 \( 1 + 18.5T + 2.80e3T^{2} \)
59 \( 1 + 85.8T + 3.48e3T^{2} \)
61 \( 1 - 82.2iT - 3.72e3T^{2} \)
67 \( 1 - 26.4iT - 4.48e3T^{2} \)
71 \( 1 - 29.1iT - 5.04e3T^{2} \)
73 \( 1 - 18.8T + 5.32e3T^{2} \)
79 \( 1 - 83.9T + 6.24e3T^{2} \)
83 \( 1 - 127.T + 6.88e3T^{2} \)
89 \( 1 + 78.3iT - 7.92e3T^{2} \)
97 \( 1 - 20.6T + 9.40e3T^{2} \)
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

−8.968188912784004214761048579293, −8.166726384997670242793120990010, −7.15322917554490887507054862501, −6.55285690154566713297489866838, −5.70761132104291987734526447834, −4.73460357897975599603740139103, −4.01758079660143822052472900799, −2.83897762148615395412226869667, −1.81923215113842758846104427235, −0.66555430374677921574694711144, 1.18267067516567491575820004394, 1.96179285653903548976763646667, 3.32961780378443521050578365283, 4.13795689377839041794258731208, 5.03592670572174150438409948014, 6.19984708409254176422501877965, 6.38455953482289760996919593976, 7.75949528846299791412036061161, 8.141214415883483684680166660520, 9.286106684895749128675493110262

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