↑

Properties

Label 2-3060-1020.539-c0-0-3
Degree $2$
Conductor $3060$
Sign $0.0705 + 0.997i$
Analytic cond. $1.52713$
Root an. cond. $1.23577$
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  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.923 − 0.382i)5-s + (−0.923 − 0.382i)8-s − i·10-s + (−1.30 + 1.30i)13-s − i·16-s − i·17-s + (0.923 − 0.382i)20-s + (0.707 + 0.707i)25-s + (−1.70 − 0.707i)26-s + (−1.08 − 0.216i)29-s + (0.923 − 0.382i)32-s + (0.923 − 0.382i)34-s + (−1.08 − 1.63i)37-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.923 − 0.382i)5-s + (−0.923 − 0.382i)8-s − i·10-s + (−1.30 + 1.30i)13-s − i·16-s − i·17-s + (0.923 − 0.382i)20-s + (0.707 + 0.707i)25-s + (−1.70 − 0.707i)26-s + (−1.08 − 0.216i)29-s + (0.923 − 0.382i)32-s + (0.923 − 0.382i)34-s + (−1.08 − 1.63i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0705 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0705 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.0705 + 0.997i$
Analytic conductor: \(1.52713\)
Root analytic conductor: \(1.23577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :0),\ 0.0705 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1977020001\)
\(L(\frac12)\) \(\approx\) \(0.1977020001\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
3 \( 1 \)
5 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + iT \)
good7 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
97 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
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

−8.630236859992749269971755020124, −7.73467127881093839154683935200, −7.16978902700648625670061269145, −6.75294003413688011174961135833, −5.44910516932755168525901363517, −4.94568963162904303718705739271, −4.15345656943169994746886005422, −3.46269810102443951897547254292, −2.18292402519365948740609674941, −0.10445967130165771643345075849, 1.54568439434762437266663169695, 2.87408361486546142778320323868, 3.29088067713319102404187555382, 4.33614030454744644763886375685, 4.98439411021608231401862842842, 5.86325299220567445308313160777, 6.78717993459072568101819142504, 7.79397972681141014073616708526, 8.235334469328090609838756387053, 9.215971545516241031542366678249

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