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Properties

Label 2-540-180.79-c0-0-0
Degree $2$
Conductor $540$
Sign $-0.342 - 0.939i$
Analytic cond. $0.269495$
Root an. cond. $0.519129$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·28-s + (−0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s − 0.999·35-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·28-s + (−0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s − 0.999·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(0.269495\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6788475171\)
\(L(\frac12)\) \(\approx\) \(0.6788475171\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.97017062096837622727523026908, −10.27620481528115088537215819704, −9.329979014909354021152153654827, −8.832143153405060182665829442173, −7.52519517997436693382247180723, −6.83451333307522758265591328945, −5.89319377979588682701216326736, −5.24649080765084844197287196045, −3.52664835806404417048596800116, −2.07320807731151890144516204983, 1.06818385127464349816642610461, 2.59372777403914185558863819939, 3.96557977926722032178687934316, 4.79864218464574690665485036118, 6.20544926634159153008711595560, 7.41238559431710544597151247096, 8.319425501913797546385383074319, 9.232752757954356762436883107694, 9.872053017151147102207854555176, 10.62593273313949135147153047916

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