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Properties

Label 2-6e3-72.29-c2-0-4
Degree $2$
Conductor $216$
Sign $-0.999 + 0.0288i$
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  + (0.405 + 1.95i)2-s + (−3.67 + 1.58i)4-s + (1.53 + 2.65i)5-s + (−0.720 + 1.24i)7-s + (−4.60 − 6.54i)8-s + (−4.57 + 4.07i)10-s + (−8.82 + 15.2i)11-s + (−8.00 + 4.61i)13-s + (−2.73 − 0.904i)14-s + (10.9 − 11.6i)16-s + 4.69i·17-s − 16.8i·19-s + (−9.83 − 7.30i)20-s + (−33.5 − 11.0i)22-s + (−33.8 + 19.5i)23-s + ⋯
L(s)  = 1  + (0.202 + 0.979i)2-s + (−0.917 + 0.397i)4-s + (0.306 + 0.530i)5-s + (−0.102 + 0.178i)7-s + (−0.575 − 0.817i)8-s + (−0.457 + 0.407i)10-s + (−0.802 + 1.39i)11-s + (−0.615 + 0.355i)13-s + (−0.195 − 0.0646i)14-s + (0.684 − 0.729i)16-s + 0.276i·17-s − 0.888i·19-s + (−0.491 − 0.365i)20-s + (−1.52 − 0.503i)22-s + (−1.47 + 0.849i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.999 + 0.0288i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.999 + 0.0288i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0158403 - 1.09934i\)
\(L(\frac12)\) \(\approx\) \(0.0158403 - 1.09934i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.405 - 1.95i)T \)
3 \( 1 \)
good5 \( 1 + (-1.53 - 2.65i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.720 - 1.24i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.82 - 15.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (8.00 - 4.61i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 4.69iT - 289T^{2} \)
19 \( 1 + 16.8iT - 361T^{2} \)
23 \( 1 + (33.8 - 19.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-7.60 + 13.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-24.3 - 42.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 14.1iT - 1.36e3T^{2} \)
41 \( 1 + (8.78 - 5.07i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.3 + 11.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.8 - 21.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 71.2T + 2.80e3T^{2} \)
59 \( 1 + (34.9 + 60.6i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-89.7 - 51.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-10.5 + 6.07i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 112. iT - 5.04e3T^{2} \)
73 \( 1 - 84.0T + 5.32e3T^{2} \)
79 \( 1 + (22.9 - 39.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (25.7 - 44.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 105. iT - 7.92e3T^{2} \)
97 \( 1 + (-19.4 + 33.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.68976421557736986026853756770, −11.92630793070529310897547969977, −10.25781586884163287217589949892, −9.696286768473284744276687271653, −8.410604256285980826387627475990, −7.33807863571680670018145975666, −6.59960720285366471652691918866, −5.33996288213330047242950003778, −4.30876610389418395158910997427, −2.54779542096759765749837836246, 0.55974514180167280998188451224, 2.40807424018570222725529767137, 3.76230945127933098290750274808, 5.14171608528280582382340000598, 5.99228564059004587211033681315, 7.936926763561092270255904526361, 8.752252020326138600713764392368, 9.943051683574939690412801316431, 10.56167480245815467419995149100, 11.68937566659745177525235119981

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