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Properties

Label 2-6e3-216.203-c1-0-16
Degree $2$
Conductor $216$
Sign $-0.513 + 0.858i$
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  + (−1.39 + 0.245i)2-s + (−1.67 − 0.440i)3-s + (1.87 − 0.684i)4-s + (2.44 + 0.202i)6-s + (−2.44 + 1.41i)8-s + (2.61 + 1.47i)9-s + (−3.02 − 3.60i)11-s + (−3.44 + 0.317i)12-s + (3.06 − 2.57i)16-s + (−4.26 − 2.46i)17-s + (−4 − 1.41i)18-s + (−3.49 − 6.05i)19-s + (5.10 + 4.28i)22-s + (4.72 − 1.28i)24-s + (−0.868 − 4.92i)25-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.967 − 0.254i)3-s + (0.939 − 0.342i)4-s + (0.996 + 0.0825i)6-s + (−0.866 + 0.500i)8-s + (0.870 + 0.491i)9-s + (−0.912 − 1.08i)11-s + (−0.995 + 0.0917i)12-s + (0.766 − 0.642i)16-s + (−1.03 − 0.597i)17-s + (−0.942 − 0.333i)18-s + (−0.801 − 1.38i)19-s + (1.08 + 0.912i)22-s + (0.964 − 0.263i)24-s + (−0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.513 + 0.858i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.513 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159823 - 0.281760i\)
\(L(\frac12)\) \(\approx\) \(0.159823 - 0.281760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.245i)T \)
3 \( 1 + (1.67 + 0.440i)T \)
good5 \( 1 + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (3.02 + 3.60i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (4.26 + 2.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.49 + 6.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.73 - 0.835i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (7.73 - 6.48i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-9.44 + 11.2i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.78 - 10.0i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.53 + 14.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-16.7 + 2.95i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (11.0 - 6.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.75 + 2.30i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.54234037903626015712840421502, −11.03252943567168564143364177584, −10.22806676431006766336719816063, −8.983140493416642396498959445151, −7.998599439135632094827056676925, −6.86700038089901538940001374743, −6.06354590113207859322633945831, −4.84176314501776027370848504449, −2.49810244578805269487478744429, −0.39649575247039052752499485777, 1.94742179623034283521018067488, 4.02172426637476945469042903849, 5.53497297934343912886277746703, 6.67909588286006232528077035652, 7.62056868364561752087581212234, 8.831748443786384806388968615500, 10.06495498923230451382916150883, 10.47609147724296341687893054490, 11.47623309991098944908591204822, 12.41716817578833814550644948165

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