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Properties

Label 2-6e3-27.4-c1-0-5
Degree $2$
Conductor $216$
Sign $0.721 + 0.692i$
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 + 0.329i)3-s + (0.198 − 1.12i)5-s + (0.914 − 0.767i)7-s + (2.78 − 1.11i)9-s + (−0.411 − 2.33i)11-s + (4.28 − 1.55i)13-s + (0.0328 + 1.97i)15-s + (2.15 − 3.72i)17-s + (−0.315 − 0.547i)19-s + (−1.30 + 1.60i)21-s + (1.05 + 0.887i)23-s + (3.47 + 1.26i)25-s + (−4.36 + 2.81i)27-s + (−9.61 − 3.50i)29-s + (3.28 + 2.75i)31-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)3-s + (0.0885 − 0.502i)5-s + (0.345 − 0.289i)7-s + (0.927 − 0.373i)9-s + (−0.124 − 0.704i)11-s + (1.18 − 0.432i)13-s + (0.00848 + 0.509i)15-s + (0.522 − 0.904i)17-s + (−0.0724 − 0.125i)19-s + (−0.284 + 0.350i)21-s + (0.220 + 0.185i)23-s + (0.695 + 0.253i)25-s + (−0.840 + 0.542i)27-s + (−1.78 − 0.650i)29-s + (0.589 + 0.494i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889047 - 0.357470i\)
\(L(\frac12)\) \(\approx\) \(0.889047 - 0.357470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.329i)T \)
good5 \( 1 + (-0.198 + 1.12i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.914 + 0.767i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.411 + 2.33i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-4.28 + 1.55i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.15 + 3.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.315 + 0.547i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.05 - 0.887i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (9.61 + 3.50i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.28 - 2.75i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (4.23 - 7.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.42 - 1.24i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.34 + 7.62i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.89 + 1.58i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 1.65T + 53T^{2} \)
59 \( 1 + (2.05 - 11.6i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-1.11 + 0.932i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-14.0 + 5.11i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.74 + 6.48i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.0 + 4.75i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.64 - 1.69i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-4.93 - 8.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.74 + 9.90i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.03690573650368204044544599598, −11.21562991208974359365039604227, −10.50122710387281548120869422573, −9.337870086185444601473700548347, −8.248494200062733129602797558113, −6.99644458397342364566461336423, −5.77557077962285470834400966197, −4.98655759200052203395230553181, −3.58595204936099985386958287263, −1.05387510892785686815090781691, 1.77136981183924307910174830649, 3.87128422701574955395927217659, 5.25090369885132287982243189231, 6.25548407057582494671517533809, 7.17719824586641804515225774074, 8.383060909888301678696042031138, 9.729450094804541056701742408937, 10.77400543317572827177635386165, 11.30446888380568448350576445871, 12.43485964757986106065510283319

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