↑

Properties

Label 2-1020-1020.959-c0-0-2
Degree $2$
Conductor $1020$
Sign $0.684 + 0.729i$
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  + (−0.555 − 0.831i)2-s + (0.831 + 0.555i)3-s + (−0.382 + 0.923i)4-s + (−0.195 − 0.980i)5-s − i·6-s + (0.980 − 0.195i)8-s + (0.382 + 0.923i)9-s + (−0.707 + 0.707i)10-s + (−0.831 + 0.555i)12-s + (0.382 − 0.923i)15-s + (−0.707 − 0.707i)16-s + (0.555 − 0.831i)17-s + (0.555 − 0.831i)18-s + (0.707 + 0.292i)19-s + (0.980 + 0.195i)20-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (0.831 + 0.555i)3-s + (−0.382 + 0.923i)4-s + (−0.195 − 0.980i)5-s − i·6-s + (0.980 − 0.195i)8-s + (0.382 + 0.923i)9-s + (−0.707 + 0.707i)10-s + (−0.831 + 0.555i)12-s + (0.382 − 0.923i)15-s + (−0.707 − 0.707i)16-s + (0.555 − 0.831i)17-s + (0.555 − 0.831i)18-s + (0.707 + 0.292i)19-s + (0.980 + 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.684 + 0.729i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ 0.684 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9938408227\)
\(L(\frac12)\) \(\approx\) \(0.9938408227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.555 + 0.831i)T \)
3 \( 1 + (-0.831 - 0.555i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
17 \( 1 + (-0.555 + 0.831i)T \)
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 \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
53 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2} \)
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

−9.875797276484329675348893204753, −9.218827444494808689915925223899, −8.672915734511981959347686551202, −7.88800591576228711326354781931, −7.15838110424204599538458016226, −5.27713841357596722203610384839, −4.58721709049832966126516697225, −3.58861749088713380231812670006, −2.69327570477689414155649533067, −1.30866386034064000342885965218, 1.50372496417796516090769864050, 2.90414418262204518980355236962, 3.92404312526901189223598570644, 5.37297955732020718745462861246, 6.39243084934949642481286594936, 7.05954331678276289284016634016, 7.72449268001297937109259213368, 8.366954827360770676460575142118, 9.379599488726743721675978714032, 9.917686056674926505365422432649

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