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Properties

Label 2-6e3-216.211-c2-0-45
Degree $2$
Conductor $216$
Sign $0.706 + 0.708i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.53 + 1.28i)2-s + (1.90 − 2.31i)3-s + (0.694 − 3.93i)4-s + (0.0553 + 5.99i)6-s + (4.00 + 6.92i)8-s + (−1.72 − 8.83i)9-s + (13.4 + 4.88i)11-s + (−7.79 − 9.12i)12-s + (−15.0 − 5.47i)16-s + (10.1 − 17.5i)17-s + (14.0 + 11.3i)18-s + (−18.8 − 32.6i)19-s + (−26.8 + 9.76i)22-s + (23.6 + 3.94i)24-s + (19.1 − 16.0i)25-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.635 − 0.771i)3-s + (0.173 − 0.984i)4-s + (0.00922 + 0.999i)6-s + (0.500 + 0.866i)8-s + (−0.191 − 0.981i)9-s + (1.22 + 0.444i)11-s + (−0.649 − 0.760i)12-s + (−0.939 − 0.342i)16-s + (0.596 − 1.03i)17-s + (0.777 + 0.628i)18-s + (−0.993 − 1.72i)19-s + (−1.22 + 0.444i)22-s + (0.986 + 0.164i)24-s + (0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.706 + 0.708i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.706 + 0.708i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21440 - 0.504022i\)
\(L(\frac12)\) \(\approx\) \(1.21440 - 0.504022i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.53 - 1.28i)T \)
3 \( 1 + (-1.90 + 2.31i)T \)
good5 \( 1 + (-19.1 + 16.0i)T^{2} \)
7 \( 1 + (46.0 - 16.7i)T^{2} \)
11 \( 1 + (-13.4 - 4.88i)T + (92.6 + 77.7i)T^{2} \)
13 \( 1 + (-29.3 - 166. i)T^{2} \)
17 \( 1 + (-10.1 + 17.5i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (18.8 + 32.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (497. + 180. i)T^{2} \)
29 \( 1 + (-146. + 828. i)T^{2} \)
31 \( 1 + (903. + 328. i)T^{2} \)
37 \( 1 + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-50.8 - 42.7i)T + (291. + 1.65e3i)T^{2} \)
43 \( 1 + (-41.1 - 14.9i)T + (1.41e3 + 1.18e3i)T^{2} \)
47 \( 1 + (2.07e3 - 755. i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (45.1 - 16.4i)T + (2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (81.3 + 68.2i)T + (779. + 4.42e3i)T^{2} \)
71 \( 1 + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-65.2 - 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (26.7 - 22.4i)T + (1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (-80.5 - 139. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-172. - 62.7i)T + (7.20e3 + 6.04e3i)T^{2} \)
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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

−11.93531645606585087219625759669, −10.94743180240874961588639497137, −9.459637710464650842443779545476, −9.060244446060688845168570361917, −7.916230230755022706531055447532, −6.95839309218344914918648172356, −6.30836788981705837223205693337, −4.60426970468366408689557426260, −2.57000834742299159479509153500, −0.962662162274854992753350357774, 1.71579074800554632054834364879, 3.39799843259334004474728572376, 4.16286924812775611090859001328, 6.07839097585222597773553829381, 7.63693173955778304173244664199, 8.566456640833349794303987877018, 9.242473647406152309787120232130, 10.29139566063814625161040308764, 10.89776458006390576825492173001, 12.07143793994452486997408481135

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