↑

Properties

Label 2-540-180.119-c1-0-25
Degree $2$
Conductor $540$
Sign $0.995 - 0.0926i$
Analytic cond. $4.31192$
Root an. cond. $2.07651$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 0.0449i)2-s + (1.99 + 0.126i)4-s + (1.91 + 1.14i)5-s + (1.44 − 2.50i)7-s + (2.81 + 0.269i)8-s + (2.65 + 1.71i)10-s + (−0.395 + 0.684i)11-s + (−4.04 + 2.33i)13-s + (2.15 − 3.47i)14-s + (3.96 + 0.506i)16-s − 5.89·17-s − 4.55i·19-s + (3.68 + 2.53i)20-s + (−0.589 + 0.950i)22-s + (1.15 − 0.666i)23-s + ⋯
L(s)  = 1  + (0.999 + 0.0317i)2-s + (0.997 + 0.0634i)4-s + (0.857 + 0.513i)5-s + (0.546 − 0.945i)7-s + (0.995 + 0.0951i)8-s + (0.841 + 0.540i)10-s + (−0.119 + 0.206i)11-s + (−1.12 + 0.647i)13-s + (0.575 − 0.927i)14-s + (0.991 + 0.126i)16-s − 1.42·17-s − 1.04i·19-s + (0.823 + 0.567i)20-s + (−0.125 + 0.202i)22-s + (0.240 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.995 - 0.0926i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.995 - 0.0926i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.04877 + 0.141581i\)
\(L(\frac12)\) \(\approx\) \(3.04877 + 0.141581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0449i)T \)
3 \( 1 \)
5 \( 1 + (-1.91 - 1.14i)T \)
good7 \( 1 + (-1.44 + 2.50i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.395 - 0.684i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.04 - 2.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 + (-1.15 + 0.666i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.29 + 1.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.26 + 1.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.44iT - 37T^{2} \)
41 \( 1 + (-5.91 + 3.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.95 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.60 + 1.50i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.70T + 53T^{2} \)
59 \( 1 + (5.29 + 9.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.869 - 1.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.76 + 6.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.88T + 71T^{2} \)
73 \( 1 + 6.50iT - 73T^{2} \)
79 \( 1 + (3.30 + 1.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.33 + 4.81i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.00741iT - 89T^{2} \)
97 \( 1 + (-5.74 - 3.31i)T + (48.5 + 84.0i)T^{2} \)
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

−11.03618567791829230063813555922, −10.17645944382840395395908282356, −9.267152491797051652550503018288, −7.76843776086804964846729059852, −6.92435925294021011865663250539, −6.37348518714210984089939225522, −4.91695157430408502293878651134, −4.44830159780333062124797219720, −2.87116729305930001226295230447, −1.86893422851487591776591693575, 1.86509538080956450431033357567, 2.72110658528253429546587202991, 4.34983742164122399963675880021, 5.32042682199867782182792358559, 5.79046062608588510325838191828, 6.95666956810972003382356473305, 8.103725572819912568335901583216, 9.035635847191370639120763059264, 10.07835520777914374729210150080, 10.93117215882395461981475778194

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