↑

Properties

Label 2-1020-1020.839-c0-0-3
Degree $2$
Conductor $1020$
Sign $0.714 + 0.699i$
Analytic cond. $0.509046$
Root an. cond. $0.713474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.980 + 0.195i)2-s + (−0.195 − 0.980i)3-s + (0.923 + 0.382i)4-s + (0.555 − 0.831i)5-s − i·6-s + (0.831 + 0.555i)8-s + (−0.923 + 0.382i)9-s + (0.707 − 0.707i)10-s + (0.195 − 0.980i)12-s + (−0.923 − 0.382i)15-s + (0.707 + 0.707i)16-s + (−0.980 + 0.195i)17-s + (−0.980 + 0.195i)18-s + (−0.707 + 1.70i)19-s + (0.831 − 0.555i)20-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)2-s + (−0.195 − 0.980i)3-s + (0.923 + 0.382i)4-s + (0.555 − 0.831i)5-s − i·6-s + (0.831 + 0.555i)8-s + (−0.923 + 0.382i)9-s + (0.707 − 0.707i)10-s + (0.195 − 0.980i)12-s + (−0.923 − 0.382i)15-s + (0.707 + 0.707i)16-s + (−0.980 + 0.195i)17-s + (−0.980 + 0.195i)18-s + (−0.707 + 1.70i)19-s + (0.831 − 0.555i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ 0.714 + 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.807509612\)
\(L(\frac12)\) \(\approx\) \(1.807509612\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.980 - 0.195i)T \)
3 \( 1 + (0.195 + 0.980i)T \)
5 \( 1 + (-0.555 + 0.831i)T \)
17 \( 1 + (0.980 - 0.195i)T \)
good7 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.275 + 1.38i)T + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (1.08 - 0.216i)T + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
53 \( 1 + (0.360 + 0.149i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.382 + 0.923i)T^{2} \)
79 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (1.81 + 0.750i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.382 - 0.923i)T^{2} \)
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

−10.33693067871105712294341374846, −8.883771511097382786508958546092, −8.283992660436138577731954678156, −7.36632875833089230604240239280, −6.36158724820210639361805893589, −5.91163955467139033102758511483, −4.95463803359148568739743879779, −3.99496511590026719772555926259, −2.48397349363565068338920974242, −1.62471948237383795379034278580, 2.21523844245786414118111678529, 3.10990551138146135250890262820, 4.07915179586384571771313299592, 5.00872283751293764482065622433, 5.78559314892990651113702492435, 6.64413130259411966961425930154, 7.37763823658275292031677483731, 8.968702707232646602305268605706, 9.569829530711258041553374462767, 10.56039370663431927881389845253

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