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Properties

Label 2-6e3-72.67-c2-0-6
Degree $2$
Conductor $216$
Sign $-0.426 - 0.904i$
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  + (1.58 + 1.21i)2-s + (1.02 + 3.86i)4-s + (4.40 + 2.54i)5-s + (−10.9 + 6.32i)7-s + (−3.09 + 7.37i)8-s + (3.88 + 9.40i)10-s + (−4.51 − 7.81i)11-s + (9.68 + 5.59i)13-s + (−25.0 − 3.33i)14-s + (−13.9 + 7.92i)16-s + 19.2·17-s + 14.2·19-s + (−5.32 + 19.6i)20-s + (2.38 − 17.8i)22-s + (−4.28 − 2.47i)23-s + ⋯
L(s)  = 1  + (0.792 + 0.609i)2-s + (0.256 + 0.966i)4-s + (0.881 + 0.508i)5-s + (−1.56 + 0.904i)7-s + (−0.386 + 0.922i)8-s + (0.388 + 0.940i)10-s + (−0.410 − 0.710i)11-s + (0.744 + 0.430i)13-s + (−1.79 − 0.238i)14-s + (−0.868 + 0.495i)16-s + 1.13·17-s + 0.749·19-s + (−0.266 + 0.982i)20-s + (0.108 − 0.813i)22-s + (−0.186 − 0.107i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.426 - 0.904i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.426 - 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21991 + 1.92502i\)
\(L(\frac12)\) \(\approx\) \(1.21991 + 1.92502i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.58 - 1.21i)T \)
3 \( 1 \)
good5 \( 1 + (-4.40 - 2.54i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (10.9 - 6.32i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.51 + 7.81i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.68 - 5.59i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 19.2T + 289T^{2} \)
19 \( 1 - 14.2T + 361T^{2} \)
23 \( 1 + (4.28 + 2.47i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (7.55 - 4.36i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-33.9 - 19.6i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 19.9iT - 1.36e3T^{2} \)
41 \( 1 + (-17.3 + 30.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (3.02 + 5.24i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-52.8 + 30.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 6.53iT - 2.80e3T^{2} \)
59 \( 1 + (-25.0 + 43.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-0.149 + 0.0862i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (40.4 - 70.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 50.4iT - 5.04e3T^{2} \)
73 \( 1 - 24.0T + 5.32e3T^{2} \)
79 \( 1 + (-2.84 + 1.64i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (18.4 + 32.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 13.0T + 7.92e3T^{2} \)
97 \( 1 + (-44.7 - 77.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.56307789198018934634827343365, −11.78950573997936769745710915123, −10.39926552632700656042893127924, −9.410626523074342709815896931305, −8.382493594626167733883466578841, −6.91524055098093505170081697019, −6.05444563742027798681862426529, −5.52649786562407316330016951894, −3.51309298512360301052205261336, −2.66699150428171959933673964993, 1.04934921537627479701123563087, 2.88242870718724983755056996794, 4.02892501503507961683697947048, 5.48783761549866365404814280844, 6.26115100126419053947483874680, 7.50299423522202660316155026179, 9.459033590938721811615119636940, 9.871709520613303157056526155240, 10.66991379600923900172213283457, 12.09144769029543461157370017141

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