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Properties

Label 2-6e3-216.11-c1-0-9
Degree $2$
Conductor $216$
Sign $-0.0429 - 0.999i$
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  + (−0.909 + 1.08i)2-s + (1.42 + 0.986i)3-s + (−0.347 − 1.96i)4-s + (−2.36 + 0.644i)6-s + (2.44 + 1.41i)8-s + (1.05 + 2.80i)9-s + (1.85 + 5.09i)11-s + (1.44 − 3.14i)12-s + (−3.75 + 1.36i)16-s + (−0.412 + 0.238i)17-s + (−4.00 − 1.41i)18-s + (2.39 − 4.14i)19-s + (−7.19 − 2.62i)22-s + (2.09 + 4.43i)24-s + (−3.83 − 3.21i)25-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (0.821 + 0.569i)3-s + (−0.173 − 0.984i)4-s + (−0.964 + 0.263i)6-s + (0.866 + 0.500i)8-s + (0.350 + 0.936i)9-s + (0.558 + 1.53i)11-s + (0.418 − 0.908i)12-s + (−0.939 + 0.342i)16-s + (−0.100 + 0.0578i)17-s + (−0.942 − 0.333i)18-s + (0.548 − 0.949i)19-s + (−1.53 − 0.558i)22-s + (0.426 + 0.904i)24-s + (−0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0429 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0429 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.0429 - 0.999i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.0429 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790480 + 0.825167i\)
\(L(\frac12)\) \(\approx\) \(0.790480 + 0.825167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 1.08i)T \)
3 \( 1 + (-1.42 - 0.986i)T \)
good5 \( 1 + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.85 - 5.09i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.412 - 0.238i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.39 + 4.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.51 + 6.57i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-12.3 + 4.48i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-5.24 + 14.4i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.26 + 6.93i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.22 - 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (9.11 - 10.8i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (15.9 + 9.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-17.3 + 6.31i)T + (74.3 - 62.3i)T^{2} \)
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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

−12.74290170765145098406507056032, −11.32317265538837030095707206795, −10.12713713023397407399426236888, −9.555624443529732569312089553592, −8.696841939441299373403750570326, −7.60749309855814491366357321916, −6.80201119671784181037659234446, −5.18266369608364574847606582435, −4.14900221704313800123196945431, −2.10245952965868370772996521554, 1.30391266307259490174892650095, 2.95183407509168986513034478783, 3.90517132071301561430558104412, 6.06716646780942498225853372955, 7.42263466153222518286694391039, 8.278443062184123712885637282196, 9.067746759510947177847421648941, 9.955847301666498187823109201978, 11.23676107114745931430692663808, 11.93722132723733284963391197810

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