Properties

Label 2-6e3-72.11-c1-0-5
Degree $2$
Conductor $216$
Sign $0.855 + 0.518i$
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  + (1.12 − 0.862i)2-s + (0.511 − 1.93i)4-s + (1.60 + 2.78i)5-s + (1.82 + 1.05i)7-s + (−1.09 − 2.60i)8-s + (4.20 + 1.73i)10-s + (−3.47 − 2.00i)11-s + (−0.341 + 0.197i)13-s + (2.94 − 0.392i)14-s + (−3.47 − 1.97i)16-s − 1.20i·17-s − 1.62·19-s + (6.21 − 1.68i)20-s + (−5.62 + 0.749i)22-s + (−2.74 − 4.75i)23-s + ⋯
L(s)  = 1  + (0.792 − 0.609i)2-s + (0.255 − 0.966i)4-s + (0.719 + 1.24i)5-s + (0.688 + 0.397i)7-s + (−0.386 − 0.922i)8-s + (1.33 + 0.548i)10-s + (−1.04 − 0.605i)11-s + (−0.0948 + 0.0547i)13-s + (0.788 − 0.105i)14-s + (−0.869 − 0.494i)16-s − 0.292i·17-s − 0.372·19-s + (1.38 − 0.376i)20-s + (−1.20 + 0.159i)22-s + (−0.572 − 0.990i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.855 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89470 - 0.529346i\)
\(L(\frac12)\) \(\approx\) \(1.89470 - 0.529346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 + 0.862i)T \)
3 \( 1 \)
good5 \( 1 + (-1.60 - 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.82 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.47 + 2.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.341 - 0.197i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.20iT - 17T^{2} \)
19 \( 1 + 1.62T + 19T^{2} \)
23 \( 1 + (2.74 + 4.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.95 - 5.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.34 + 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (-1.23 + 0.715i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.21 - 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.792 - 1.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 + (-2.29 + 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.18 + 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.60 + 4.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.69T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + (-1.53 - 0.886i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.30 - 0.755i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.19883677807832864871933717886, −11.14516344180139896258843120480, −10.59404400242735637965169848793, −9.745220451169809004624846941034, −8.255691767984928416600951979076, −6.79109683545752707774743806822, −5.87923603800714144416519818603, −4.83309964051970877593037703554, −3.11087842226862977687247479677, −2.17788314410282264070507524753, 2.08784552186305106219218933182, 4.15779471407644360437760518384, 5.10025993369227482620920628679, 5.85746704742874363973750593970, 7.43370933310331235205427069731, 8.184929101223090810046542315474, 9.280554339616210428563314160049, 10.51526722722042845185272523773, 11.78448320695217600722749541162, 12.72535648296736909160206129784

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