Properties

Label 2-6e3-72.5-c2-0-14
Degree $2$
Conductor $216$
Sign $-0.599 + 0.800i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 − 1.62i)2-s + (−1.26 + 3.79i)4-s + (2.90 − 5.03i)5-s + (−0.363 − 0.629i)7-s + (7.63 − 2.38i)8-s + (−11.5 + 1.17i)10-s + (−2.03 − 3.51i)11-s + (13.3 + 7.70i)13-s + (−0.596 + 1.32i)14-s + (−12.8 − 9.59i)16-s − 11.6i·17-s − 35.6i·19-s + (15.4 + 17.4i)20-s + (−3.33 + 7.40i)22-s + (−26.0 − 15.0i)23-s + ⋯
L(s)  = 1  + (−0.584 − 0.811i)2-s + (−0.316 + 0.948i)4-s + (0.581 − 1.00i)5-s + (−0.0519 − 0.0899i)7-s + (0.954 − 0.298i)8-s + (−1.15 + 0.117i)10-s + (−0.184 − 0.319i)11-s + (1.02 + 0.592i)13-s + (−0.0425 + 0.0946i)14-s + (−0.800 − 0.599i)16-s − 0.682i·17-s − 1.87i·19-s + (0.771 + 0.870i)20-s + (−0.151 + 0.336i)22-s + (−1.13 − 0.654i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.599 + 0.800i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.497465 - 0.993901i\)
\(L(\frac12)\) \(\approx\) \(0.497465 - 0.993901i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 1.62i)T \)
3 \( 1 \)
good5 \( 1 + (-2.90 + 5.03i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (0.363 + 0.629i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (2.03 + 3.51i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.3 - 7.70i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 11.6iT - 289T^{2} \)
19 \( 1 + 35.6iT - 361T^{2} \)
23 \( 1 + (26.0 + 15.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (16.7 + 28.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (17.8 - 30.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 24.7iT - 1.36e3T^{2} \)
41 \( 1 + (-49.7 - 28.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (6.79 - 3.92i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-4.39 + 2.53i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 28.1T + 2.80e3T^{2} \)
59 \( 1 + (-19.8 + 34.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-33.9 + 19.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-63.9 - 36.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 88.5iT - 5.04e3T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 + (-35.5 - 61.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (18.1 + 31.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 121. iT - 7.92e3T^{2} \)
97 \( 1 + (25.3 + 43.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.57141348614207195159096105700, −10.88377869619725762563032548609, −9.588978402213142265905763473248, −9.027942840488755588957331589871, −8.155700229814566440215039410607, −6.74363642000236605894830202245, −5.19894110239963152158985966943, −4.00124147314437530302918114936, −2.32203999617172060039994383774, −0.78189072657586187149196414040, 1.81350737667846781330493706323, 3.76828686342755697782782086890, 5.72682584596671766427501241184, 6.16061368673457425179949453576, 7.46841372121446169702963487490, 8.281182806857781533515653089353, 9.571188413075356746651256627428, 10.35747644372237781216704128130, 10.99363225051382451460394145120, 12.55853655546102911033633021581

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