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Properties

Label 2-540-180.59-c1-0-5
Degree $2$
Conductor $540$
Sign $-0.656 - 0.754i$
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  + (−0.0443 + 1.41i)2-s + (−1.99 − 0.125i)4-s + (1.69 + 1.46i)5-s + (−0.667 − 1.15i)7-s + (0.265 − 2.81i)8-s + (−2.14 + 2.32i)10-s + (2.18 + 3.78i)11-s + (3.56 + 2.05i)13-s + (1.66 − 0.892i)14-s + (3.96 + 0.500i)16-s − 6.45·17-s + 5.84i·19-s + (−3.19 − 3.12i)20-s + (−5.45 + 2.92i)22-s + (0.0875 + 0.0505i)23-s + ⋯
L(s)  = 1  + (−0.0313 + 0.999i)2-s + (−0.998 − 0.0626i)4-s + (0.756 + 0.653i)5-s + (−0.252 − 0.436i)7-s + (0.0939 − 0.995i)8-s + (−0.677 + 0.735i)10-s + (0.659 + 1.14i)11-s + (0.987 + 0.570i)13-s + (0.444 − 0.238i)14-s + (0.992 + 0.125i)16-s − 1.56·17-s + 1.34i·19-s + (−0.714 − 0.699i)20-s + (−1.16 + 0.623i)22-s + (0.0182 + 0.0105i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $-0.656 - 0.754i$
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: \((2,\ 540,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566885 + 1.24515i\)
\(L(\frac12)\) \(\approx\) \(0.566885 + 1.24515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0443 - 1.41i)T \)
3 \( 1 \)
5 \( 1 + (-1.69 - 1.46i)T \)
good7 \( 1 + (0.667 + 1.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.56 - 2.05i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.45T + 17T^{2} \)
19 \( 1 - 5.84iT - 19T^{2} \)
23 \( 1 + (-0.0875 - 0.0505i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.53 + 2.61i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.24iT - 37T^{2} \)
41 \( 1 + (3.50 + 2.02i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.92 - 3.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.00 - 1.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.77T + 53T^{2} \)
59 \( 1 + (1.37 - 2.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.04 - 1.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.216 + 0.374i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.41T + 71T^{2} \)
73 \( 1 + 7.28iT - 73T^{2} \)
79 \( 1 + (2.58 - 1.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.6 + 6.70i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.97iT - 89T^{2} \)
97 \( 1 + (-2.08 + 1.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.88892719580250261166721681311, −10.01453571051488200268446541894, −9.365505061131789224167458852845, −8.463350300406283711521922672133, −7.24347260577555964440601744890, −6.58872756848053724421815353721, −5.98764779370373418828283291415, −4.57695943933502260868688601112, −3.70433678608919670901793470287, −1.79167470606825312992990758054, 0.869017520698175881412438735180, 2.31576622001543966053186398706, 3.48808398696369667250852754908, 4.72206355477619598795532627459, 5.68867745991594014307398779397, 6.61063831932635353434483063049, 8.528761173920638528718376085674, 8.785352211036262275619991049542, 9.505850955335764417077607038822, 10.75949445005767985215335283132

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