Properties

Label 2-6e3-27.22-c1-0-2
Degree $2$
Conductor $216$
Sign $0.961 - 0.275i$
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  + (−1.70 + 0.327i)3-s + (1.34 − 0.491i)5-s + (0.111 + 0.629i)7-s + (2.78 − 1.11i)9-s + (2.56 + 0.933i)11-s + (3.51 + 2.95i)13-s + (−2.13 + 1.27i)15-s + (1.37 − 2.38i)17-s + (2.25 + 3.89i)19-s + (−0.394 − 1.03i)21-s + (0.868 − 4.92i)23-s + (−2.25 + 1.88i)25-s + (−4.37 + 2.80i)27-s + (2.34 − 1.96i)29-s + (0.510 − 2.89i)31-s + ⋯
L(s)  = 1  + (−0.981 + 0.188i)3-s + (0.603 − 0.219i)5-s + (0.0419 + 0.237i)7-s + (0.928 − 0.371i)9-s + (0.773 + 0.281i)11-s + (0.976 + 0.819i)13-s + (−0.551 + 0.329i)15-s + (0.333 − 0.577i)17-s + (0.516 + 0.894i)19-s + (−0.0861 − 0.225i)21-s + (0.181 − 1.02i)23-s + (−0.450 + 0.377i)25-s + (−0.841 + 0.539i)27-s + (0.434 − 0.364i)29-s + (0.0916 − 0.519i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05959 + 0.148954i\)
\(L(\frac12)\) \(\approx\) \(1.05959 + 0.148954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.70 - 0.327i)T \)
good5 \( 1 + (-1.34 + 0.491i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.111 - 0.629i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.56 - 0.933i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.51 - 2.95i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.37 + 2.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.25 - 3.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.868 + 4.92i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.34 + 1.96i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.510 + 2.89i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.60 - 6.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.31 + 6.97i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (9.16 + 3.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.382 - 2.17i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 + (-11.1 + 4.07i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.62 - 9.24i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.12 + 4.29i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (6.61 - 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.02 - 3.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.4 + 8.80i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.98 + 2.50i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (3.94 + 6.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.0 + 5.83i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.00639627658391421314153573049, −11.66840584517566300733496307987, −10.36330429426531700057135449616, −9.616485946238068906603990569956, −8.583037129267876084546299057087, −6.96197936120546591907473969133, −6.12827777431081622986032901576, −5.11769659784332860017786115181, −3.85477178016134980235157921419, −1.54848806377506592133663210107, 1.35169556401301589752584834669, 3.54133097830349173687456090887, 5.13283724967992723363663237889, 6.08668361431808377583686140969, 6.92443377862894434204131244291, 8.232045769962380033549556811679, 9.569855141275057539930636988213, 10.49749671505103166772161737177, 11.26280859847822468652312412768, 12.16934700574226853839458817319

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