Properties

Label 2-540-15.14-c2-0-5
Degree $2$
Conductor $540$
Sign $0.872 - 0.488i$
Analytic cond. $14.7139$
Root an. cond. $3.83587$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.36 + 2.44i)5-s − 2.79i·7-s − 18.1i·11-s + 23.0i·13-s + 5.72·17-s + 23.1·19-s + 0.271·23-s + (13.0 − 21.2i)25-s + 39.7i·29-s + 47.3·31-s + (6.82 + 12.2i)35-s + 34.8i·37-s + 13.2i·41-s + 46.7i·43-s − 40.9·47-s + ⋯
L(s)  = 1  + (−0.872 + 0.488i)5-s − 0.399i·7-s − 1.64i·11-s + 1.77i·13-s + 0.336·17-s + 1.22·19-s + 0.0118·23-s + (0.523 − 0.851i)25-s + 1.37i·29-s + 1.52·31-s + (0.194 + 0.348i)35-s + 0.942i·37-s + 0.323i·41-s + 1.08i·43-s − 0.870·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(14.7139\)
Root analytic conductor: \(3.83587\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1),\ 0.872 - 0.488i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.458599510\)
\(L(\frac12)\) \(\approx\) \(1.458599510\)
\(L(2)\) 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 \( 1 + (4.36 - 2.44i)T \)
good7 \( 1 + 2.79iT - 49T^{2} \)
11 \( 1 + 18.1iT - 121T^{2} \)
13 \( 1 - 23.0iT - 169T^{2} \)
17 \( 1 - 5.72T + 289T^{2} \)
19 \( 1 - 23.1T + 361T^{2} \)
23 \( 1 - 0.271T + 529T^{2} \)
29 \( 1 - 39.7iT - 841T^{2} \)
31 \( 1 - 47.3T + 961T^{2} \)
37 \( 1 - 34.8iT - 1.36e3T^{2} \)
41 \( 1 - 13.2iT - 1.68e3T^{2} \)
43 \( 1 - 46.7iT - 1.84e3T^{2} \)
47 \( 1 + 40.9T + 2.20e3T^{2} \)
53 \( 1 - 91.3T + 2.80e3T^{2} \)
59 \( 1 + 78.8iT - 3.48e3T^{2} \)
61 \( 1 - 31.1T + 3.72e3T^{2} \)
67 \( 1 - 6.91iT - 4.48e3T^{2} \)
71 \( 1 - 81.5iT - 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 - 63.5T + 6.24e3T^{2} \)
83 \( 1 + 0.284T + 6.88e3T^{2} \)
89 \( 1 - 28.5iT - 7.92e3T^{2} \)
97 \( 1 - 92.9iT - 9.40e3T^{2} \)
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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

−10.87818185207270490287712907947, −9.845431488513289114133920409654, −8.787426955882121379425837123471, −8.035579620599367863590679104455, −7.01275670838954465020833305254, −6.31335588900175068848086073253, −4.93343453075699766950193652077, −3.80559388466228488940765137014, −2.98460700410702451579507870099, −1.02290069279573665269526890177, 0.76760870114311455137911986603, 2.54622887259097882223733260228, 3.82051151121896728239863977875, 4.91146452031773010622554800347, 5.70298091042920946086717120353, 7.24486505499314037530679437588, 7.76278599163881387647142556416, 8.661414007539616987532208570800, 9.783249010540094298896499855454, 10.35960695316802119939884389210

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