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Properties

Label 2-6e3-72.67-c2-0-16
Degree $2$
Conductor $216$
Sign $0.817 + 0.575i$
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  + (1.95 − 0.434i)2-s + (3.62 − 1.69i)4-s + (0.0166 + 0.00958i)5-s + (4.07 − 2.35i)7-s + (6.33 − 4.88i)8-s + (0.0365 + 0.0114i)10-s + (−2.84 − 4.93i)11-s + (10.0 + 5.80i)13-s + (6.92 − 6.35i)14-s + (10.2 − 12.2i)16-s + 0.376·17-s − 15.0·19-s + (0.0763 + 0.00653i)20-s + (−7.70 − 8.39i)22-s + (39.1 + 22.6i)23-s + ⋯
L(s)  = 1  + (0.976 − 0.217i)2-s + (0.905 − 0.424i)4-s + (0.00332 + 0.00191i)5-s + (0.581 − 0.335i)7-s + (0.791 − 0.611i)8-s + (0.00365 + 0.00114i)10-s + (−0.259 − 0.448i)11-s + (0.773 + 0.446i)13-s + (0.494 − 0.454i)14-s + (0.639 − 0.768i)16-s + 0.0221·17-s − 0.792·19-s + (0.00381 + 0.000326i)20-s + (−0.350 − 0.381i)22-s + (1.70 + 0.983i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.817 + 0.575i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.79371 - 0.884393i\)
\(L(\frac12)\) \(\approx\) \(2.79371 - 0.884393i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 + 0.434i)T \)
3 \( 1 \)
good5 \( 1 + (-0.0166 - 0.00958i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.07 + 2.35i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.84 + 4.93i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-10.0 - 5.80i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 0.376T + 289T^{2} \)
19 \( 1 + 15.0T + 361T^{2} \)
23 \( 1 + (-39.1 - 22.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (32.0 - 18.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (26.3 + 15.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 53.4iT - 1.36e3T^{2} \)
41 \( 1 + (29.0 - 50.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (23.0 + 39.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-34.2 + 19.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 0.989iT - 2.80e3T^{2} \)
59 \( 1 + (29.4 - 50.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (75.1 - 43.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (34.1 - 59.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 42.3iT - 5.04e3T^{2} \)
73 \( 1 - 26.6T + 5.32e3T^{2} \)
79 \( 1 + (-121. + 69.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (40.9 + 71.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 42.6T + 7.92e3T^{2} \)
97 \( 1 + (-55.9 - 96.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.94453125746713835374012673143, −11.14556883188856647377502492132, −10.50592088982082513610014376303, −9.067595067668913970992782835534, −7.78997799583408960812624127684, −6.68819870242952411853681848622, −5.57170841612238384290815012243, −4.45428786572166445137093239148, −3.26841479304736523537649228657, −1.56946914797012323773316802909, 2.01396277612468349177364221554, 3.53802206273682605438008712237, 4.84484559213302990947302325115, 5.76064680089964042108376894169, 6.97986123841030679669327296892, 7.985404755486428927027366631263, 9.076281038113977562039127635260, 10.75709994462185154498660024270, 11.18393349616694621470324549172, 12.50338647301973678710060953098

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