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Properties

Label 2-540-45.34-c1-0-3
Degree $2$
Conductor $540$
Sign $0.802 + 0.597i$
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  + (−1.99 + 1.00i)5-s + (−1.11 − 0.644i)7-s + (2.54 − 4.41i)11-s + (3.09 − 1.78i)13-s + 0.895i·17-s + 5.34·19-s + (4.38 − 2.53i)23-s + (2.96 − 4.02i)25-s + (1.5 − 2.59i)29-s + (3.29 + 5.71i)31-s + (2.87 + 0.159i)35-s + 7.24i·37-s + (−3.92 − 6.79i)41-s + (−9.46 − 5.46i)43-s + (−2.57 − 1.48i)47-s + ⋯
L(s)  = 1  + (−0.892 + 0.451i)5-s + (−0.421 − 0.243i)7-s + (0.767 − 1.32i)11-s + (0.857 − 0.495i)13-s + 0.217i·17-s + 1.22·19-s + (0.914 − 0.528i)23-s + (0.592 − 0.805i)25-s + (0.278 − 0.482i)29-s + (0.592 + 1.02i)31-s + (0.486 + 0.0269i)35-s + 1.19i·37-s + (−0.612 − 1.06i)41-s + (−1.44 − 0.833i)43-s + (−0.375 − 0.216i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.802 + 0.597i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.802 + 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17559 - 0.389483i\)
\(L(\frac12)\) \(\approx\) \(1.17559 - 0.389483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1.99 - 1.00i)T \)
good7 \( 1 + (1.11 + 0.644i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.54 + 4.41i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.09 + 1.78i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.895iT - 17T^{2} \)
19 \( 1 - 5.34T + 19T^{2} \)
23 \( 1 + (-4.38 + 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.29 - 5.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.24iT - 37T^{2} \)
41 \( 1 + (3.92 + 6.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.46 + 5.46i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.57 + 1.48i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.78iT - 53T^{2} \)
59 \( 1 + (-2.87 - 4.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.17 - 3.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.39 + 4.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.34T + 71T^{2} \)
73 \( 1 + 9.34iT - 73T^{2} \)
79 \( 1 + (0.370 - 0.642i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.97 - 4.60i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + (2.99 + 1.72i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.81528310879246530378381440112, −10.00546589947223788177962132649, −8.688235789971587269614611642641, −8.246124800570079194220784964643, −6.97073723214347617831581086813, −6.35604522731236312162984920164, −5.09441796989205440170891735362, −3.60419002841348214536964051392, −3.21447885433940350995374465049, −0.869216522313488507855641659268, 1.36371355206149548880943962466, 3.19437374519020495837010020498, 4.22138713708211864870769531543, 5.14376187266042436316961363196, 6.51606912558658617376840424443, 7.28342790298644918825629688839, 8.237883363926565212855224249109, 9.296546444916526649129143266163, 9.720121539234109299039025208134, 11.22133453652685743590618312280

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