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Properties

Label 2-6e3-72.13-c1-0-8
Degree $2$
Conductor $216$
Sign $-0.933 - 0.359i$
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  + (−0.820 − 1.15i)2-s + (−0.654 + 1.88i)4-s + (−1.97 − 1.14i)5-s + (−0.907 − 1.57i)7-s + (2.71 − 0.795i)8-s + (0.306 + 3.21i)10-s + (−4.24 + 2.44i)11-s + (−4.00 − 2.31i)13-s + (−1.06 + 2.33i)14-s + (−3.14 − 2.47i)16-s − 1.92·17-s + 2.12i·19-s + (3.44 − 2.98i)20-s + (6.30 + 2.87i)22-s + (1.15 − 2.00i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.814i)2-s + (−0.327 + 0.944i)4-s + (−0.883 − 0.510i)5-s + (−0.343 − 0.594i)7-s + (0.959 − 0.281i)8-s + (0.0968 + 1.01i)10-s + (−1.27 + 0.738i)11-s + (−1.11 − 0.641i)13-s + (−0.285 + 0.624i)14-s + (−0.785 − 0.618i)16-s − 0.467·17-s + 0.488i·19-s + (0.771 − 0.667i)20-s + (1.34 + 0.613i)22-s + (0.241 − 0.418i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.933 - 0.359i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0432053 + 0.232585i\)
\(L(\frac12)\) \(\approx\) \(0.0432053 + 0.232585i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.820 + 1.15i)T \)
3 \( 1 \)
good5 \( 1 + (1.97 + 1.14i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.907 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.24 - 2.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.00 + 2.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 - 2.12iT - 19T^{2} \)
23 \( 1 + (-1.15 + 2.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.16 + 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.65 - 4.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.98iT - 37T^{2} \)
41 \( 1 + (-2.36 + 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.02 - 3.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.95iT - 53T^{2} \)
59 \( 1 + (-3.05 - 1.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.71 - 0.991i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.72 - 4.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (4.97 + 8.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.12 + 1.80i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 + (-6.99 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.86586236610939113504459530677, −10.56655938448393779726388594909, −10.12429556056137588657273344798, −8.857349822886173038365560604987, −7.79800925107036395633258647588, −7.24408358688003509157871583403, −5.02387700397642519750228932324, −3.99952181393726488108461680756, −2.52935640703144750636400486476, −0.22735496729930546521910322768, 2.74904789831606887210176424309, 4.62069493372382132835561745758, 5.79971954218215620874620689117, 7.01948256460463291397360487302, 7.77335893817623211914734177740, 8.779880487750386560501754435362, 9.777952017246394902184047695306, 10.85931619725685103496820546789, 11.65625751659639797458822361157, 12.97665741228564896699887343772

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