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Properties

Label 2-6e3-8.3-c2-0-7
Degree $2$
Conductor $216$
Sign $0.145 - 0.989i$
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  + (−1.77 + 0.914i)2-s + (2.32 − 3.25i)4-s + 3.96i·5-s − 3.99i·7-s + (−1.16 + 7.91i)8-s + (−3.62 − 7.05i)10-s + 9.34·11-s + 10.5i·13-s + (3.65 + 7.11i)14-s + (−5.16 − 15.1i)16-s − 7.91·17-s + 21.7·19-s + (12.8 + 9.22i)20-s + (−16.6 + 8.54i)22-s + 34.7i·23-s + ⋯
L(s)  = 1  + (−0.889 + 0.457i)2-s + (0.581 − 0.813i)4-s + 0.792i·5-s − 0.571i·7-s + (−0.145 + 0.989i)8-s + (−0.362 − 0.705i)10-s + 0.849·11-s + 0.815i·13-s + (0.261 + 0.508i)14-s + (−0.322 − 0.946i)16-s − 0.465·17-s + 1.14·19-s + (0.644 + 0.461i)20-s + (−0.755 + 0.388i)22-s + 1.51i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.145 - 0.989i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.145 - 0.989i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.758602 + 0.655185i\)
\(L(\frac12)\) \(\approx\) \(0.758602 + 0.655185i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.77 - 0.914i)T \)
3 \( 1 \)
good5 \( 1 - 3.96iT - 25T^{2} \)
7 \( 1 + 3.99iT - 49T^{2} \)
11 \( 1 - 9.34T + 121T^{2} \)
13 \( 1 - 10.5iT - 169T^{2} \)
17 \( 1 + 7.91T + 289T^{2} \)
19 \( 1 - 21.7T + 361T^{2} \)
23 \( 1 - 34.7iT - 529T^{2} \)
29 \( 1 - 8.37iT - 841T^{2} \)
31 \( 1 - 49.4iT - 961T^{2} \)
37 \( 1 - 56.8iT - 1.36e3T^{2} \)
41 \( 1 + 53.6T + 1.68e3T^{2} \)
43 \( 1 + 6.33T + 1.84e3T^{2} \)
47 \( 1 - 3.84iT - 2.20e3T^{2} \)
53 \( 1 + 102. iT - 2.80e3T^{2} \)
59 \( 1 - 116.T + 3.48e3T^{2} \)
61 \( 1 + 57.4iT - 3.72e3T^{2} \)
67 \( 1 - 13.1T + 4.48e3T^{2} \)
71 \( 1 + 48.4iT - 5.04e3T^{2} \)
73 \( 1 - 59.0T + 5.32e3T^{2} \)
79 \( 1 + 42.7iT - 6.24e3T^{2} \)
83 \( 1 + 0.0660T + 6.88e3T^{2} \)
89 \( 1 + 98.4T + 7.92e3T^{2} \)
97 \( 1 + 158.T + 9.40e3T^{2} \)
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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

−11.82607074847159228545861092750, −11.25191260353692402047881124246, −10.18805615834003284954582371878, −9.422931476429854876735265681078, −8.350143554301366195412762765551, −6.95892410154433450913262737374, −6.78125227171717716443497000844, −5.16125716450800221924842460412, −3.38278980736481541409334619694, −1.49260256604953595574936317663, 0.825717444749708391755955723813, 2.51542781508704133589476907473, 4.10052092873560537603506968889, 5.66559670810654970141271105439, 6.99993286067749730258648921701, 8.230524489430445854163205936587, 8.941223857991950757980511597053, 9.748631245912741804793659668894, 10.87715618499236638067438440046, 11.89425787027371105463421191124

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