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Properties

Label 2-6e3-27.7-c1-0-3
Degree $2$
Conductor $216$
Sign $0.999 - 0.0427i$
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  + (−0.158 − 1.72i)3-s + (0.536 + 3.04i)5-s + (3.96 + 3.32i)7-s + (−2.94 + 0.547i)9-s + (0.729 − 4.13i)11-s + (0.702 + 0.255i)13-s + (5.16 − 1.40i)15-s + (−0.749 − 1.29i)17-s + (2.08 − 3.60i)19-s + (5.11 − 7.37i)21-s + (−3.23 + 2.71i)23-s + (−4.26 + 1.55i)25-s + (1.41 + 5.00i)27-s + (−4.93 + 1.79i)29-s + (−0.393 + 0.330i)31-s + ⋯
L(s)  = 1  + (−0.0916 − 0.995i)3-s + (0.239 + 1.36i)5-s + (1.49 + 1.25i)7-s + (−0.983 + 0.182i)9-s + (0.219 − 1.24i)11-s + (0.194 + 0.0708i)13-s + (1.33 − 0.363i)15-s + (−0.181 − 0.314i)17-s + (0.477 − 0.827i)19-s + (1.11 − 1.60i)21-s + (−0.674 + 0.565i)23-s + (−0.853 + 0.310i)25-s + (0.271 + 0.962i)27-s + (−0.915 + 0.333i)29-s + (−0.0706 + 0.0592i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.999 - 0.0427i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.999 - 0.0427i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34242 + 0.0286892i\)
\(L(\frac12)\) \(\approx\) \(1.34242 + 0.0286892i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.158 + 1.72i)T \)
good5 \( 1 + (-0.536 - 3.04i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-3.96 - 3.32i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.729 + 4.13i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.702 - 0.255i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.749 + 1.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.08 + 3.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.23 - 2.71i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.93 - 1.79i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.393 - 0.330i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-1.08 - 1.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.58 - 0.939i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.79 + 10.1i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.66 + 4.75i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 1.54T + 53T^{2} \)
59 \( 1 + (1.00 + 5.68i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.80 + 6.55i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.94 + 1.43i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.71 + 4.69i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.08 + 1.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (15.0 - 5.47i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-2.10 + 0.766i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (1.83 - 3.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.650 + 3.69i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.03228212168490944090154702975, −11.34315755214829078651368642193, −10.92597747695414976880510127054, −9.122101650989675351380623301742, −8.238413351297243886884562084875, −7.27429269460138368740008394356, −6.14054690631737475433457788232, −5.35388973106814259723983663506, −3.07200419831311765306653002198, −1.93973332149500946125078593351, 1.51564995611301973944945685201, 4.23464798379508148782959365212, 4.53947752468536481246680638099, 5.70164542489027715444629463470, 7.57711464075279722148737784649, 8.399224000967378711525916083877, 9.489828430730637209557911785624, 10.29740376759766297147149411941, 11.26329187178507655468273416041, 12.19977232928248726046085235209

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