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Properties

Label 2-6e3-8.5-c1-0-5
Degree $2$
Conductor $216$
Sign $-0.148 - 0.988i$
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  + (0.767 + 1.18i)2-s + (−0.822 + 1.82i)4-s + 0.841i·5-s + 2.64·7-s + (−2.79 + 0.420i)8-s + (−0.999 + 0.645i)10-s + 3.91i·11-s − 0.645i·13-s + (2.02 + 3.14i)14-s + (−2.64 − 2.99i)16-s − 5.59·17-s − 4.29i·19-s + (−1.53 − 0.692i)20-s + (−4.64 + 3.00i)22-s + 8.66·23-s + ⋯
L(s)  = 1  + (0.542 + 0.840i)2-s + (−0.411 + 0.911i)4-s + 0.376i·5-s + 0.999·7-s + (−0.988 + 0.148i)8-s + (−0.316 + 0.204i)10-s + 1.17i·11-s − 0.179i·13-s + (0.542 + 0.840i)14-s + (−0.661 − 0.749i)16-s − 1.35·17-s − 0.984i·19-s + (−0.343 − 0.154i)20-s + (−0.990 + 0.639i)22-s + 1.80·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.148 - 0.988i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.148 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02428 + 1.18993i\)
\(L(\frac12)\) \(\approx\) \(1.02428 + 1.18993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.767 - 1.18i)T \)
3 \( 1 \)
good5 \( 1 - 0.841iT - 5T^{2} \)
7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 3.91iT - 11T^{2} \)
13 \( 1 + 0.645iT - 13T^{2} \)
17 \( 1 + 5.59T + 17T^{2} \)
19 \( 1 + 4.29iT - 19T^{2} \)
23 \( 1 - 8.66T + 23T^{2} \)
29 \( 1 + 7.82iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 6.64iT - 37T^{2} \)
41 \( 1 + 6.13T + 41T^{2} \)
43 \( 1 - 7.29iT - 43T^{2} \)
47 \( 1 - 2.52T + 47T^{2} \)
53 \( 1 + 7.82iT - 53T^{2} \)
59 \( 1 + 7.27iT - 59T^{2} \)
61 \( 1 - 13.9iT - 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.58T + 73T^{2} \)
79 \( 1 + 3.35T + 79T^{2} \)
83 \( 1 - 9.50iT - 83T^{2} \)
89 \( 1 - 5.59T + 89T^{2} \)
97 \( 1 + 8.29T + 97T^{2} \)
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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

−12.87811252458209336944219133198, −11.66506679407841556867920647177, −10.89576110249425903270943947476, −9.383387451416342370024617698544, −8.439809901233853388099151819369, −7.30279820599429863364463296021, −6.64395192333326584916404006561, −5.06980373568661328811280952404, −4.39988119399264794708372031902, −2.57940208782148185530571697162, 1.39264991324237704461846571164, 3.11128409266332721575431850417, 4.55771632226627770947947903093, 5.38585485837981594725335896407, 6.74030515119704575054822205626, 8.503398962910524119355428671825, 8.993338928106170271508068784763, 10.57105674362845392084205540006, 11.09848483878279041696049766329, 11.97773715415822483293696563592

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