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Properties

Label 2-540-20.7-c1-0-27
Degree $2$
Conductor $540$
Sign $0.635 + 0.772i$
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  + (−1.22 + 0.698i)2-s + (1.02 − 1.71i)4-s + (0.252 − 2.22i)5-s + (1.10 − 1.10i)7-s + (−0.0578 + 2.82i)8-s + (1.24 + 2.90i)10-s + 3.20i·11-s + (1.72 − 1.72i)13-s + (−0.588 + 2.13i)14-s + (−1.90 − 3.51i)16-s + (−1.47 − 1.47i)17-s + 6.17·19-s + (−3.55 − 2.70i)20-s + (−2.24 − 3.94i)22-s + (−3.97 − 3.97i)23-s + ⋯
L(s)  = 1  + (−0.869 + 0.494i)2-s + (0.511 − 0.859i)4-s + (0.113 − 0.993i)5-s + (0.418 − 0.418i)7-s + (−0.0204 + 0.999i)8-s + (0.392 + 0.919i)10-s + 0.967i·11-s + (0.478 − 0.478i)13-s + (−0.157 + 0.571i)14-s + (−0.476 − 0.879i)16-s + (−0.357 − 0.357i)17-s + 1.41·19-s + (−0.795 − 0.605i)20-s + (−0.477 − 0.840i)22-s + (−0.829 − 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.887447 - 0.419056i\)
\(L(\frac12)\) \(\approx\) \(0.887447 - 0.419056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.698i)T \)
3 \( 1 \)
5 \( 1 + (-0.252 + 2.22i)T \)
good7 \( 1 + (-1.10 + 1.10i)T - 7iT^{2} \)
11 \( 1 - 3.20iT - 11T^{2} \)
13 \( 1 + (-1.72 + 1.72i)T - 13iT^{2} \)
17 \( 1 + (1.47 + 1.47i)T + 17iT^{2} \)
19 \( 1 - 6.17T + 19T^{2} \)
23 \( 1 + (3.97 + 3.97i)T + 23iT^{2} \)
29 \( 1 + 7.67iT - 29T^{2} \)
31 \( 1 + 1.41iT - 31T^{2} \)
37 \( 1 + (3.62 + 3.62i)T + 37iT^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + (3.53 + 3.53i)T + 43iT^{2} \)
47 \( 1 + (2.03 - 2.03i)T - 47iT^{2} \)
53 \( 1 + (-5.27 + 5.27i)T - 53iT^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 + 3.24T + 61T^{2} \)
67 \( 1 + (10.9 - 10.9i)T - 67iT^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (-9.81 + 9.81i)T - 73iT^{2} \)
79 \( 1 - 0.893T + 79T^{2} \)
83 \( 1 + (0.944 + 0.944i)T + 83iT^{2} \)
89 \( 1 + 8.66iT - 89T^{2} \)
97 \( 1 + (-11.8 - 11.8i)T + 97iT^{2} \)
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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

−10.39846917886105902408867611756, −9.688934268075507408815948805417, −8.937990457381861190866839644040, −7.945965517232811170950192063090, −7.42802246665979187253431072394, −6.14834824998212251133841328559, −5.21720465680389546190815480498, −4.25126797203735416545741490311, −2.17137616397626229965578243142, −0.815955499829133717146680443120, 1.56617357630144748675972159063, 2.91412361765481166007363287139, 3.77021141764244563477533296954, 5.58095860839369432956574601198, 6.58976228109308449321663843079, 7.50621048527127131862269606241, 8.393693332001789674684556286930, 9.223086187982039595639046373054, 10.10817571808145434201623370700, 11.03796355278714745774616253168

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