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Properties

Label 2-540-36.11-c1-0-14
Degree $2$
Conductor $540$
Sign $0.987 - 0.159i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 + 0.228i)2-s + (1.89 + 0.638i)4-s + (0.866 − 0.5i)5-s + (−1.04 − 0.604i)7-s + (2.49 + 1.32i)8-s + (1.32 − 0.499i)10-s + (1.52 − 2.63i)11-s + (2.53 + 4.39i)13-s + (−1.32 − 1.08i)14-s + (3.18 + 2.42i)16-s − 2.23i·17-s + 2.53i·19-s + (1.96 − 0.394i)20-s + (2.72 − 3.32i)22-s + (−3.48 − 6.02i)23-s + ⋯
L(s)  = 1  + (0.986 + 0.161i)2-s + (0.947 + 0.319i)4-s + (0.387 − 0.223i)5-s + (−0.395 − 0.228i)7-s + (0.883 + 0.468i)8-s + (0.418 − 0.157i)10-s + (0.458 − 0.794i)11-s + (0.704 + 1.21i)13-s + (−0.353 − 0.289i)14-s + (0.795 + 0.605i)16-s − 0.543i·17-s + 0.582i·19-s + (0.438 − 0.0881i)20-s + (0.581 − 0.709i)22-s + (−0.725 − 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.987 - 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.81908 + 0.226410i\)
\(L(\frac12)\) \(\approx\) \(2.81908 + 0.226410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.228i)T \)
3 \( 1 \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (1.04 + 0.604i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.52 + 2.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.53 - 4.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 - 2.53iT - 19T^{2} \)
23 \( 1 + (3.48 + 6.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.84 - 2.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.83 - 1.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (3.70 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.74 - 2.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.15 - 8.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.42iT - 53T^{2} \)
59 \( 1 + (4.46 + 7.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.32 + 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.4 - 7.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.49T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 + (-3.26 - 1.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.286 - 0.495i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.40iT - 89T^{2} \)
97 \( 1 + (-5.82 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.02134759852004813818547695811, −10.13904082332617053845340681396, −8.972293102282007333586938041305, −8.141953009966862339323172416379, −6.68207890001558464486737108499, −6.40512918814453897930996808465, −5.20468454980249245363370821221, −4.14611497746566808543584531548, −3.19284411197862697543790302530, −1.68331929007880971824943886323, 1.70326088207121485607913461368, 3.02858880103276919913539173701, 3.97982873305563307372644328570, 5.27760922193629525041747002753, 6.03269729087591338965438613437, 6.90446560800969431561565556685, 7.905451397370390924021903977107, 9.213831414712777426454009561094, 10.21035229272497599156118492539, 10.75743229626809900200632174674

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