Properties

Label 2-1020-1020.539-c0-0-0
Degree $2$
Conductor $1020$
Sign $-0.939 - 0.341i$
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  + (−0.195 + 0.980i)2-s + (−0.555 − 0.831i)3-s + (−0.923 − 0.382i)4-s + (−0.980 − 0.195i)5-s + (0.923 − 0.382i)6-s + (0.555 − 0.831i)8-s + (−0.382 + 0.923i)9-s + (0.382 − 0.923i)10-s + (0.195 + 0.980i)12-s + (0.382 + 0.923i)15-s + (0.707 + 0.707i)16-s + (−0.831 + 0.555i)17-s + (−0.831 − 0.555i)18-s + (−0.707 + 0.292i)19-s + (0.831 + 0.555i)20-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)2-s + (−0.555 − 0.831i)3-s + (−0.923 − 0.382i)4-s + (−0.980 − 0.195i)5-s + (0.923 − 0.382i)6-s + (0.555 − 0.831i)8-s + (−0.382 + 0.923i)9-s + (0.382 − 0.923i)10-s + (0.195 + 0.980i)12-s + (0.382 + 0.923i)15-s + (0.707 + 0.707i)16-s + (−0.831 + 0.555i)17-s + (−0.831 − 0.555i)18-s + (−0.707 + 0.292i)19-s + (0.831 + 0.555i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.939 - 0.341i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ -0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2007522275\)
\(L(\frac12)\) \(\approx\) \(0.2007522275\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.195 - 0.980i)T \)
3 \( 1 + (0.555 + 0.831i)T \)
5 \( 1 + (0.980 + 0.195i)T \)
17 \( 1 + (0.831 - 0.555i)T \)
good7 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.785 - 1.17i)T + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.275 - 0.275i)T - iT^{2} \)
53 \( 1 + (-0.425 - 1.02i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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

−10.69116283257771374655000852561, −9.420171794958841637113316146729, −8.491788441920866261458490176961, −7.921891124314710482105688596626, −7.18238072665353756646505448809, −6.45259923837375032015508659923, −5.56600050359970285189920591623, −4.65075396149465432620644008841, −3.64530506494762668126937678153, −1.62994738404096938322916247199, 0.21669656063834619983073197022, 2.40579435070972329083106504992, 3.61379322224330567630790825518, 4.28361741139901757862588359760, 5.01895012307769842000564684276, 6.30127934533145608779108588647, 7.41259406526613279962414300237, 8.498164230137040473550419847299, 9.046850948507717696220462979967, 10.02454124336541756594670031459

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