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Properties

Label 2-6e3-24.11-c1-0-11
Degree $2$
Conductor $216$
Sign $0.166 + 0.986i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.637 + 1.26i)2-s + (−1.18 − 1.61i)4-s − 3.42·5-s − 0.505i·7-s + (2.78 − 0.469i)8-s + (2.18 − 4.32i)10-s − 3.31i·11-s − 5.43i·13-s + (0.637 + 0.322i)14-s + (−1.18 + 3.82i)16-s − 1.58i·17-s − 6.74·19-s + (4.06 + 5.51i)20-s + (4.18 + 2.11i)22-s − 4.30·23-s + ⋯
L(s)  = 1  + (−0.451 + 0.892i)2-s + (−0.593 − 0.805i)4-s − 1.53·5-s − 0.191i·7-s + (0.986 − 0.166i)8-s + (0.691 − 1.36i)10-s − 1.00i·11-s − 1.50i·13-s + (0.170 + 0.0861i)14-s + (−0.296 + 0.955i)16-s − 0.384i·17-s − 1.54·19-s + (0.908 + 1.23i)20-s + (0.892 + 0.451i)22-s − 0.897·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.166 + 0.986i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284830 - 0.240858i\)
\(L(\frac12)\) \(\approx\) \(0.284830 - 0.240858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.637 - 1.26i)T \)
3 \( 1 \)
good5 \( 1 + 3.42T + 5T^{2} \)
7 \( 1 + 0.505iT - 7T^{2} \)
11 \( 1 + 3.31iT - 11T^{2} \)
13 \( 1 + 5.43iT - 13T^{2} \)
17 \( 1 + 1.58iT - 17T^{2} \)
19 \( 1 + 6.74T + 19T^{2} \)
23 \( 1 + 4.30T + 23T^{2} \)
29 \( 1 - 2.55T + 29T^{2} \)
31 \( 1 - 4.92iT - 31T^{2} \)
37 \( 1 - 7.45iT - 37T^{2} \)
41 \( 1 + 8.51iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 5.10T + 47T^{2} \)
53 \( 1 - 0.875T + 53T^{2} \)
59 \( 1 - 6.63iT - 59T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 + 3.74T + 73T^{2} \)
79 \( 1 + 6.44iT - 79T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + 3.75iT - 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−12.07886816290252410184887874420, −10.90810279788714745652000956870, −10.25331203324146140217971959423, −8.559875989374947650796134389764, −8.225997832273341853320226050715, −7.23132110583580894221987164625, −6.05814771477405651219254701758, −4.73540364698453789065378995821, −3.49297725429386736978068400014, −0.36242347159115108842919282640, 2.13579790967784039335202917291, 3.96819751148054241069790214088, 4.44718136422141314262410056445, 6.74392701888681370851375104423, 7.81704106208072574168272012777, 8.610464446331037098736122444899, 9.654156640543881196110178738006, 10.78422633091367484912355381315, 11.65629613713910926542498261805, 12.19401494537977664382376346493

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