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Properties

Label 2-6e3-9.5-c2-0-2
Degree $2$
Conductor $216$
Sign $0.668 - 0.743i$
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  + (8.20 + 4.73i)5-s + (1.05 + 1.83i)7-s + (−13.7 + 7.91i)11-s + (4.70 − 8.14i)13-s + 11.6i·17-s + 12.9·19-s + (5.27 + 3.04i)23-s + (32.4 + 56.1i)25-s + (24.7 − 14.2i)29-s + (8.75 − 15.1i)31-s + 20.0i·35-s − 15.6·37-s + (−14.8 − 8.54i)41-s + (−21.7 − 37.6i)43-s + (−20.6 + 11.9i)47-s + ⋯
L(s)  = 1  + (1.64 + 0.947i)5-s + (0.150 + 0.261i)7-s + (−1.24 + 0.719i)11-s + (0.361 − 0.626i)13-s + 0.682i·17-s + 0.682·19-s + (0.229 + 0.132i)23-s + (1.29 + 2.24i)25-s + (0.854 − 0.493i)29-s + (0.282 − 0.489i)31-s + 0.572i·35-s − 0.422·37-s + (−0.361 − 0.208i)41-s + (−0.505 − 0.874i)43-s + (−0.439 + 0.253i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.76332 + 0.785788i\)
\(L(\frac12)\) \(\approx\) \(1.76332 + 0.785788i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-8.20 - 4.73i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.05 - 1.83i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (13.7 - 7.91i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.70 + 8.14i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 11.6iT - 289T^{2} \)
19 \( 1 - 12.9T + 361T^{2} \)
23 \( 1 + (-5.27 - 3.04i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-24.7 + 14.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-8.75 + 15.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 15.6T + 1.36e3T^{2} \)
41 \( 1 + (14.8 + 8.54i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (21.7 + 37.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (20.6 - 11.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + (38.5 + 22.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.86 + 3.22i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-21.0 + 36.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 120. iT - 5.04e3T^{2} \)
73 \( 1 - 5.48T + 5.32e3T^{2} \)
79 \( 1 + (-60.5 - 104. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (46.5 - 26.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 102. iT - 7.92e3T^{2} \)
97 \( 1 + (58.9 + 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.37651240557456956143179653193, −10.91880206967642013661146714422, −10.25892644167818464344046791503, −9.596055566038517798768095391779, −8.220976468236390360348599521501, −7.01963773084952254420200440643, −5.93950578053395753412210761148, −5.14000222707174394574673533527, −3.04117251112805491993060898691, −1.96278079796447593016728979215, 1.21597481944238800614590382699, 2.75672404048624938244993687737, 4.82258269911389802090620900267, 5.53527899503071940589824876616, 6.66962053196648403419767108750, 8.191176192183992254818961405895, 9.068692743531701112839834746359, 9.956750466673134938821782426745, 10.81159702153848268965194856404, 12.09531745852442290608739469976

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