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Properties

Label 2-540-20.19-c0-0-0
Degree $2$
Conductor $540$
Sign $0.866 - 0.5i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s − 0.999i·8-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)16-s + i·17-s + 1.73i·19-s + (−0.866 + 0.499i)20-s + 1.73·23-s − 25-s − 1.73i·31-s + (0.866 − 0.499i)32-s + (0.5 − 0.866i)34-s + (0.866 − 1.49i)38-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s − 0.999i·8-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)16-s + i·17-s + 1.73i·19-s + (−0.866 + 0.499i)20-s + 1.73·23-s − 25-s − 1.73i·31-s + (0.866 − 0.499i)32-s + (0.5 − 0.866i)34-s + (0.866 − 1.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(0.269495\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :0),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6047693757\)
\(L(\frac12)\) \(\approx\) \(0.6047693757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.99539355786586780019738272148, −10.22232043588161111733304096210, −9.585151724451196193814540927301, −8.413982918475851672153912430548, −7.69517622236429324940069053508, −6.76681479553440991524030923267, −5.86104103053586107272681235565, −4.01679404040763416094679008791, −3.10278667633552680431979629700, −1.82218885829353227148199695355, 1.07957635671882874792228937876, 2.77843337266637008525501289210, 4.78496276188677030228112436158, 5.28597851592803479533065110032, 6.70323847911278966464954782152, 7.34790708417681867394911794861, 8.558670641471965356455311049763, 9.034081929383756842963430812391, 9.763473041443551052644337525012, 10.93278306873702050739730493749

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