↑

Properties

Label 2-1020-1020.839-c0-0-1
Degree $2$
Conductor $1020$
Sign $0.0101 + 0.999i$
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  + (−0.831 − 0.555i)2-s + (−0.195 − 0.980i)3-s + (0.382 + 0.923i)4-s + (−0.555 + 0.831i)5-s + (−0.382 + 0.923i)6-s + (0.195 − 0.980i)8-s + (−0.923 + 0.382i)9-s + (0.923 − 0.382i)10-s + (0.831 − 0.555i)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 + (−0.980 − 0.195i)20-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (−0.195 − 0.980i)3-s + (0.382 + 0.923i)4-s + (−0.555 + 0.831i)5-s + (−0.382 + 0.923i)6-s + (0.195 − 0.980i)8-s + (−0.923 + 0.382i)9-s + (0.923 − 0.382i)10-s + (0.831 − 0.555i)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 + (−0.980 − 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0101 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.0101 + 0.999i$
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: \((2,\ 1020,\ (\ :0),\ 0.0101 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5704937219\)
\(L(\frac12)\) \(\approx\) \(0.5704937219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 + 0.555i)T \)
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

−10.12150372238207069759899330340, −9.001416854256274155470283933933, −8.252202188355742675205364939257, −7.37389526634106463306213872582, −6.99636581466910582647094996918, −6.03009764541152449469555798218, −4.52405874044843181175877852873, −3.05928400099003290080944940093, −2.53168305925684701834046944554, −0.848533627719318610801011031217, 1.29793675977589172519717675666, 3.29117716286594413260402067241, 4.33547793339858539523213656906, 5.47985225413069295442139190751, 5.78106503818461236224949151396, 7.27688107048535120796611163037, 8.058436032105047397403401691188, 8.673558135049813685496064586607, 9.633702787998083257632841539081, 9.996256876765193597938808134566

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