↑

Properties

Label 2-1008-21.2-c2-0-10
Degree $2$
Conductor $1008$
Sign $-0.0330 - 0.999i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
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  + (−0.0443 − 0.0255i)5-s + (6.98 − 0.430i)7-s + (−6.88 + 3.97i)11-s + 8.73·13-s + (−25.1 + 14.5i)17-s + (−15.5 + 26.9i)19-s + (25.1 + 14.5i)23-s + (−12.4 − 21.6i)25-s − 10.1i·29-s + (16.3 + 28.3i)31-s + (−0.320 − 0.159i)35-s + (13.7 − 23.8i)37-s − 2.97i·41-s + 7.18·43-s + (16.3 + 9.43i)47-s + ⋯
L(s)  = 1  + (−0.00886 − 0.00511i)5-s + (0.998 − 0.0615i)7-s + (−0.626 + 0.361i)11-s + 0.671·13-s + (−1.48 + 0.855i)17-s + (−0.818 + 1.41i)19-s + (1.09 + 0.632i)23-s + (−0.499 − 0.865i)25-s − 0.350i·29-s + (0.528 + 0.914i)31-s + (−0.00916 − 0.00456i)35-s + (0.371 − 0.643i)37-s − 0.0726i·41-s + 0.167·43-s + (0.347 + 0.200i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0330 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0330 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0330 - 0.999i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ -0.0330 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.535511618\)
\(L(\frac12)\) \(\approx\) \(1.535511618\)
\(L(2)\) 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 \( 1 + (-6.98 + 0.430i)T \)
good5 \( 1 + (0.0443 + 0.0255i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.88 - 3.97i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 8.73T + 169T^{2} \)
17 \( 1 + (25.1 - 14.5i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (15.5 - 26.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-25.1 - 14.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 10.1iT - 841T^{2} \)
31 \( 1 + (-16.3 - 28.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.7 + 23.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 2.97iT - 1.68e3T^{2} \)
43 \( 1 - 7.18T + 1.84e3T^{2} \)
47 \( 1 + (-16.3 - 9.43i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (86.8 - 50.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (33.9 - 19.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.31 - 9.19i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-49.6 - 86.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.0iT - 5.04e3T^{2} \)
73 \( 1 + (7.28 + 12.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.2 - 95.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 106. iT - 6.88e3T^{2} \)
89 \( 1 + (-13.3 - 7.69i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 116.T + 9.40e3T^{2} \)
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

−10.17758773555261465242458052098, −8.958285317384740657117825522464, −8.314145368277899578329457641109, −7.65147764084077578045591727597, −6.53395239279946017206576902429, −5.71652542925885201490434282615, −4.62167000507212501022972304307, −3.93342091488245443949601189522, −2.41436169505015125286856676422, −1.41699896919359969981708827921, 0.48137530347127282472594582897, 2.01757171208727911060794080349, 3.02788689888244447074342123992, 4.52777085348890197263793504445, 4.95899700587569433714991801798, 6.20069628694388702669205273036, 7.03074355090901781778342677312, 7.982499365452077328269777213445, 8.759905732593413826450744474633, 9.328642886551642670973658485423

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