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Properties

Label 2-540-180.59-c1-0-20
Degree $2$
Conductor $540$
Sign $0.124 + 0.992i$
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  + (−0.937 + 1.05i)2-s + (−0.242 − 1.98i)4-s + (−0.520 + 2.17i)5-s + (−0.550 − 0.953i)7-s + (2.32 + 1.60i)8-s + (−1.81 − 2.58i)10-s + (−2.84 − 4.92i)11-s + (−2.07 − 1.19i)13-s + (1.52 + 0.310i)14-s + (−3.88 + 0.963i)16-s − 1.29·17-s + 2.78i·19-s + (4.44 + 0.504i)20-s + (7.87 + 1.60i)22-s + (−1.82 − 1.05i)23-s + ⋯
L(s)  = 1  + (−0.662 + 0.748i)2-s + (−0.121 − 0.992i)4-s + (−0.232 + 0.972i)5-s + (−0.208 − 0.360i)7-s + (0.823 + 0.567i)8-s + (−0.574 − 0.818i)10-s + (−0.856 − 1.48i)11-s + (−0.575 − 0.332i)13-s + (0.407 + 0.0830i)14-s + (−0.970 + 0.240i)16-s − 0.313·17-s + 0.638i·19-s + (0.993 + 0.112i)20-s + (1.67 + 0.342i)22-s + (−0.380 − 0.219i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $0.124 + 0.992i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ 0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276021 - 0.243470i\)
\(L(\frac12)\) \(\approx\) \(0.276021 - 0.243470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.937 - 1.05i)T \)
3 \( 1 \)
5 \( 1 + (0.520 - 2.17i)T \)
good7 \( 1 + (0.550 + 0.953i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.84 + 4.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.07 + 1.19i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.29T + 17T^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + (1.82 + 1.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.51 + 3.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 + 1.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.51iT - 37T^{2} \)
41 \( 1 + (6.35 + 3.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.25 + 3.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.09 + 4.67i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + (3.66 - 6.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.48 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.85 - 3.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 9.86iT - 73T^{2} \)
79 \( 1 + (-0.975 + 0.563i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.98 - 5.76i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.16iT - 89T^{2} \)
97 \( 1 + (11.1 - 6.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.53859686897534546269129107534, −9.851073885820484828029734103753, −8.598760061984894547064536084657, −7.901646524491652742402524741763, −7.09901399996685728080201364636, −6.16756085410315852037528599029, −5.36854360660944599927536559861, −3.81966775331464839773675922952, −2.48023769059468781393488484951, −0.25629756877007444882224144729, 1.68397159989792678637601754689, 2.86222617666864751296721200951, 4.44197245879878932374075196615, 5.00752455386766655852443750775, 6.77975052776532774702416762801, 7.71130322197932953368477752559, 8.494206080392605946832287965807, 9.459445571251984571224733153868, 9.900318481105666278909957189217, 10.99832480852385179951504362891

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