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Properties

Label 2-540-180.59-c1-0-1
Degree $2$
Conductor $540$
Sign $-0.0690 - 0.997i$
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  + (−1.25 − 0.645i)2-s + (1.16 + 1.62i)4-s + (0.0966 + 2.23i)5-s + (1.60 + 2.78i)7-s + (−0.419 − 2.79i)8-s + (1.32 − 2.87i)10-s + (1.56 + 2.70i)11-s + (−1.59 − 0.923i)13-s + (−0.225 − 4.54i)14-s + (−1.27 + 3.79i)16-s − 5.85·17-s − 2.24i·19-s + (−3.51 + 2.76i)20-s + (−0.219 − 4.41i)22-s + (3.24 + 1.87i)23-s + ⋯
L(s)  = 1  + (−0.889 − 0.456i)2-s + (0.583 + 0.812i)4-s + (0.0432 + 0.999i)5-s + (0.608 + 1.05i)7-s + (−0.148 − 0.988i)8-s + (0.417 − 0.908i)10-s + (0.471 + 0.816i)11-s + (−0.443 − 0.256i)13-s + (−0.0602 − 1.21i)14-s + (−0.319 + 0.947i)16-s − 1.41·17-s − 0.514i·19-s + (−0.786 + 0.617i)20-s + (−0.0467 − 0.941i)22-s + (0.676 + 0.390i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0690 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0690 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $-0.0690 - 0.997i$
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: \((2,\ 540,\ (\ :1/2),\ -0.0690 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570689 + 0.611578i\)
\(L(\frac12)\) \(\approx\) \(0.570689 + 0.611578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.645i)T \)
3 \( 1 \)
5 \( 1 + (-0.0966 - 2.23i)T \)
good7 \( 1 + (-1.60 - 2.78i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.56 - 2.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.59 + 0.923i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.85T + 17T^{2} \)
19 \( 1 + 2.24iT - 19T^{2} \)
23 \( 1 + (-3.24 - 1.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.90 - 3.98i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.03 + 0.599i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.66iT - 37T^{2} \)
41 \( 1 + (-0.208 - 0.120i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.66 - 4.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.96 + 2.28i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + (3.47 - 6.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.66 - 2.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.31 - 7.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (11.6 - 6.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.07 + 0.623i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (-5.75 + 3.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.16825882393915178793719588816, −10.16358721682847020555523184278, −9.234853049860156443189104849565, −8.671172666365387106246489587781, −7.36437279005993272542596962739, −6.93317115109208289717172982420, −5.62755612565747583333113443915, −4.16171909701386717475352007629, −2.72573648196657796268005624854, −1.96558153166170924395920659578, 0.63932089623913339320281330177, 1.94193068776902385550996536196, 4.05364640401531632600101421836, 5.03237692723825704761520976971, 6.14170675545581550220067149427, 7.16844535381274484904729129896, 7.996285014232872789067351221447, 8.839437699275054419910232506408, 9.427298833166582715742572190207, 10.58510156406724437255284534777

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