Properties

Label 2-270-27.16-c1-0-3
Degree $2$
Conductor $270$
Sign $0.0891 - 0.996i$
Analytic cond. $2.15596$
Root an. cond. $1.46831$
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  + (0.766 + 0.642i)2-s + (−1.65 − 0.513i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s + (−0.936 − 1.45i)6-s + (−0.500 + 2.83i)7-s + (−0.500 + 0.866i)8-s + (2.47 + 1.70i)9-s + (0.5 + 0.866i)10-s + (0.527 − 0.191i)11-s + (0.218 − 1.71i)12-s + (−1.19 + 1.00i)13-s + (−2.20 + 1.85i)14-s + (−1.37 − 1.04i)15-s + (−0.939 + 0.342i)16-s + (3.91 + 6.78i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.954 − 0.296i)3-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s + (−0.382 − 0.594i)6-s + (−0.189 + 1.07i)7-s + (−0.176 + 0.306i)8-s + (0.823 + 0.566i)9-s + (0.158 + 0.273i)10-s + (0.159 − 0.0578i)11-s + (0.0631 − 0.495i)12-s + (−0.331 + 0.277i)13-s + (−0.590 + 0.495i)14-s + (−0.355 − 0.270i)15-s + (−0.234 + 0.0855i)16-s + (0.950 + 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0891 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0891 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.0891 - 0.996i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.0891 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945963 + 0.865047i\)
\(L(\frac12)\) \(\approx\) \(0.945963 + 0.865047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (1.65 + 0.513i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
good7 \( 1 + (0.500 - 2.83i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.527 + 0.191i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (1.19 - 1.00i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.91 - 6.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0937 + 0.162i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.230 - 1.30i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.737 - 0.618i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.56 + 8.89i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.25 + 7.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.59 + 4.69i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.71 - 0.988i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.892 + 5.06i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (9.19 + 3.34i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.18 + 6.74i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (4.70 - 3.94i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-8.25 - 14.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.61 + 2.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.20 - 4.36i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.56 - 2.98i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.46 + 2.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (17.5 - 6.38i)T + (74.3 - 62.3i)T^{2} \)
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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

−12.36620883902665790687306669936, −11.44465810097910282694349748159, −10.40163147730718930345897966006, −9.297913289267575640717080899922, −8.050455349404028648090963654678, −6.92603737145014060764263846093, −5.85885176657214636248726207156, −5.50354877526879969630464590616, −3.94919464761537237557142674555, −2.11212786964324571667116316688, 1.01575698533363063759139900560, 3.21891424992223253642519712896, 4.57446710273340837861478428274, 5.33097898275059527612045186255, 6.56135899711482626885995839747, 7.42857268117862484119644867144, 9.296071516365400147483917639337, 10.13902486217940971396152855205, 10.69392206339350099563983834189, 11.82371262659428987213352208731

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