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Properties

Label 2-540-180.119-c1-0-26
Degree $2$
Conductor $540$
Sign $0.589 + 0.807i$
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  + (1.38 − 0.288i)2-s + (1.83 − 0.799i)4-s + (−2.23 − 0.145i)5-s + (0.573 − 0.993i)7-s + (2.30 − 1.63i)8-s + (−3.13 + 0.442i)10-s + (0.629 − 1.09i)11-s + (4.39 − 2.53i)13-s + (0.507 − 1.54i)14-s + (2.72 − 2.93i)16-s + 3.29·17-s − 3.62i·19-s + (−4.20 + 1.51i)20-s + (0.556 − 1.69i)22-s + (−3.09 + 1.78i)23-s + ⋯
L(s)  = 1  + (0.978 − 0.204i)2-s + (0.916 − 0.399i)4-s + (−0.997 − 0.0650i)5-s + (0.216 − 0.375i)7-s + (0.815 − 0.578i)8-s + (−0.990 + 0.140i)10-s + (0.189 − 0.328i)11-s + (1.21 − 0.704i)13-s + (0.135 − 0.411i)14-s + (0.680 − 0.732i)16-s + 0.799·17-s − 0.831i·19-s + (−0.940 + 0.339i)20-s + (0.118 − 0.360i)22-s + (−0.646 + 0.373i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.589 + 0.807i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.589 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15938 - 1.09777i\)
\(L(\frac12)\) \(\approx\) \(2.15938 - 1.09777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.288i)T \)
3 \( 1 \)
5 \( 1 + (2.23 + 0.145i)T \)
good7 \( 1 + (-0.573 + 0.993i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.629 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.39 + 2.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
23 \( 1 + (3.09 - 1.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.184 - 0.106i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.12 - 5.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.72iT - 37T^{2} \)
41 \( 1 + (5.81 - 3.35i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.41 - 2.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.925 + 0.534i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.12T + 53T^{2} \)
59 \( 1 + (4.87 + 8.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.24 - 9.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.42 - 12.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.68T + 71T^{2} \)
73 \( 1 + 9.08iT - 73T^{2} \)
79 \( 1 + (-5.78 - 3.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.93 - 2.84i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.26iT - 89T^{2} \)
97 \( 1 + (6.69 + 3.86i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.97265950305744231975374586580, −10.20081086206668338955498636502, −8.757640344863475150110179987333, −7.82663926781159514120365292115, −7.01299146736529445553910338310, −5.91042000177066640666035926241, −4.92190145225523560723562284209, −3.80792740721838951058368486861, −3.17510058533536357182488585127, −1.21029143513842297808441592208, 1.89771627917974685579659879685, 3.56015930437224704124949177733, 4.07021656541003115866676652444, 5.34738773293127674676932452201, 6.25997822806175541806944540369, 7.29242202322047029100802778124, 8.051940378116202736218876082764, 8.950167586276419115454811761111, 10.41196324228086467307564783080, 11.22600528777895579426345361650

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