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Properties

Label 2-6e3-72.67-c2-0-15
Degree $2$
Conductor $216$
Sign $0.558 + 0.829i$
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  + (1.49 − 1.32i)2-s + (0.466 − 3.97i)4-s + (5.84 + 3.37i)5-s + (3.50 − 2.02i)7-s + (−4.58 − 6.55i)8-s + (13.2 − 2.72i)10-s + (4.13 + 7.16i)11-s + (−7.66 − 4.42i)13-s + (2.55 − 7.68i)14-s + (−15.5 − 3.70i)16-s + 28.6·17-s − 7.93·19-s + (16.1 − 21.6i)20-s + (15.7 + 5.21i)22-s + (−25.3 − 14.6i)23-s + ⋯
L(s)  = 1  + (0.747 − 0.664i)2-s + (0.116 − 0.993i)4-s + (1.16 + 0.675i)5-s + (0.501 − 0.289i)7-s + (−0.572 − 0.819i)8-s + (1.32 − 0.272i)10-s + (0.376 + 0.651i)11-s + (−0.589 − 0.340i)13-s + (0.182 − 0.549i)14-s + (−0.972 − 0.231i)16-s + 1.68·17-s − 0.417·19-s + (0.806 − 1.08i)20-s + (0.714 + 0.236i)22-s + (−1.10 − 0.635i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.558 + 0.829i$
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: \((2,\ 216,\ (\ :1),\ 0.558 + 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43529 - 1.29650i\)
\(L(\frac12)\) \(\approx\) \(2.43529 - 1.29650i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.49 + 1.32i)T \)
3 \( 1 \)
good5 \( 1 + (-5.84 - 3.37i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.50 + 2.02i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-4.13 - 7.16i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (7.66 + 4.42i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 28.6T + 289T^{2} \)
19 \( 1 + 7.93T + 361T^{2} \)
23 \( 1 + (25.3 + 14.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.7 + 9.08i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (40.8 + 23.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 13.3iT - 1.36e3T^{2} \)
41 \( 1 + (31.0 - 53.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-26.5 - 46.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (12.8 - 7.43i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-11.4 + 19.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-51.3 + 29.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-10.3 + 17.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 54.7iT - 5.04e3T^{2} \)
73 \( 1 + 27.3T + 5.32e3T^{2} \)
79 \( 1 + (62.9 - 36.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (4.34 + 7.53i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 28.2T + 7.92e3T^{2} \)
97 \( 1 + (0.200 + 0.348i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.06672254160981601997436630504, −10.94022705037003957514229908098, −10.02939087005392090378246634856, −9.634820352990028175444544950753, −7.77152058303209499990784803476, −6.46642605649053006583833302317, −5.59771328368306299153303666465, −4.37357801310072637234082261860, −2.84031765752060429335410202102, −1.62714822924448021473233600860, 1.93273539389040044995115339477, 3.69430900315232481761125416095, 5.29847721360503463234843262071, 5.63360716326407312717093410708, 6.98976360899140453332232509373, 8.243607065821414347260365442046, 9.076334154032388338849629639664, 10.19159709239065678390497598449, 11.69847295830442767767902019308, 12.37500815741203724027593203459

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