Properties

Label 2-6e3-216.43-c2-0-61
Degree $2$
Conductor $216$
Sign $-0.313 - 0.949i$
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 + (−0.0276 − 2.99i)3-s + (0.694 + 3.93i)4-s + (−3.81 + 4.63i)6-s + (4.00 − 6.92i)8-s + (−8.99 + 0.166i)9-s + (−17.9 + 6.54i)11-s + (11.7 − 2.19i)12-s + (−15.0 + 5.47i)16-s + (−11.6 − 20.2i)17-s + (14.0 + 11.3i)18-s + (−13.0 + 22.6i)19-s + (35.9 + 13.0i)22-s + (−20.8 − 11.8i)24-s + (19.1 + 16.0i)25-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.00922 − 0.999i)3-s + (0.173 + 0.984i)4-s + (−0.635 + 0.771i)6-s + (0.500 − 0.866i)8-s + (−0.999 + 0.0184i)9-s + (−1.63 + 0.595i)11-s + (0.983 − 0.182i)12-s + (−0.939 + 0.342i)16-s + (−0.686 − 1.18i)17-s + (0.777 + 0.628i)18-s + (−0.688 + 1.19i)19-s + (1.63 + 0.595i)22-s + (−0.870 − 0.491i)24-s + (0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.313 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0243438 + 0.0336624i\)
\(L(\frac12)\) \(\approx\) \(0.0243438 + 0.0336624i\)
\(L(2)\) 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 \( 1 + (0.0276 + 2.99i)T \)
good5 \( 1 + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (46.0 + 16.7i)T^{2} \)
11 \( 1 + (17.9 - 6.54i)T + (92.6 - 77.7i)T^{2} \)
13 \( 1 + (-29.3 + 166. i)T^{2} \)
17 \( 1 + (11.6 + 20.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (13.0 - 22.6i)T + (-180.5 - 312. i)T^{2} \)
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 \( 1 + (-15.3 + 12.8i)T + (291. - 1.65e3i)T^{2} \)
43 \( 1 + (61.3 - 22.3i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (2.07e3 + 755. i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (99.6 + 36.2i)T + (2.66e3 + 2.23e3i)T^{2} \)
61 \( 1 + (3.49e3 + 1.27e3i)T^{2} \)
67 \( 1 + (-97.8 + 82.1i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-43.4 + 75.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-1.08e3 - 6.14e3i)T^{2} \)
83 \( 1 + (94.2 + 79.1i)T + (1.19e3 + 6.78e3i)T^{2} \)
89 \( 1 + (7.59 - 13.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (141. - 51.5i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−11.36069655518738105510615880673, −10.55733480612833581843539469345, −9.456774722110363998472499820050, −8.280758736166731680063678042607, −7.61177270656925052270134849720, −6.62539767564521442498236217405, −4.98996620718373775835843928263, −3.02262167327000147190432551866, −1.90742848278950685962180271301, −0.02731170011889083482608911145, 2.64434344906429234565388515989, 4.55669481207710805424342592009, 5.53985093203517994362026036239, 6.64515757222249087596142046097, 8.189650914162490346360733961953, 8.654565158216194259547707502190, 9.857793893911449270463850388045, 10.72798726181744622134819891114, 11.13026109030364513556190050662, 12.90769993383719723350401219689

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