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Properties

Label 2-6e3-72.5-c2-0-19
Degree $2$
Conductor $216$
Sign $0.401 + 0.915i$
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  + (1.94 − 0.464i)2-s + (3.56 − 1.80i)4-s + (1.64 − 2.84i)5-s + (−4.94 − 8.56i)7-s + (6.09 − 5.17i)8-s + (1.87 − 6.30i)10-s + (7.26 + 12.5i)11-s + (−8.15 − 4.71i)13-s + (−13.5 − 14.3i)14-s + (9.45 − 12.9i)16-s − 16.6i·17-s + 11.8i·19-s + (0.714 − 13.1i)20-s + (19.9 + 21.0i)22-s + (27.6 + 15.9i)23-s + ⋯
L(s)  = 1  + (0.972 − 0.232i)2-s + (0.891 − 0.452i)4-s + (0.328 − 0.569i)5-s + (−0.706 − 1.22i)7-s + (0.762 − 0.647i)8-s + (0.187 − 0.630i)10-s + (0.660 + 1.14i)11-s + (−0.627 − 0.362i)13-s + (−0.971 − 1.02i)14-s + (0.590 − 0.806i)16-s − 0.977i·17-s + 0.622i·19-s + (0.0357 − 0.656i)20-s + (0.907 + 0.958i)22-s + (1.20 + 0.693i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.401 + 0.915i$
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: \((2,\ 216,\ (\ :1),\ 0.401 + 0.915i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.31620 - 1.51367i\)
\(L(\frac12)\) \(\approx\) \(2.31620 - 1.51367i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 + 0.464i)T \)
3 \( 1 \)
good5 \( 1 + (-1.64 + 2.84i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (4.94 + 8.56i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.26 - 12.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (8.15 + 4.71i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 16.6iT - 289T^{2} \)
19 \( 1 - 11.8iT - 361T^{2} \)
23 \( 1 + (-27.6 - 15.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.0 - 22.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (8.69 - 15.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 40.7iT - 1.36e3T^{2} \)
41 \( 1 + (-6.97 - 4.02i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (50.5 - 29.1i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (0.0690 - 0.0398i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 10.9T + 2.80e3T^{2} \)
59 \( 1 + (53.8 - 93.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (46.2 - 26.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-69.3 - 40.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 16.8iT - 5.04e3T^{2} \)
73 \( 1 - 30.6T + 5.32e3T^{2} \)
79 \( 1 + (-46.7 - 81.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-7.74 - 13.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 12.2iT - 7.92e3T^{2} \)
97 \( 1 + (91.1 + 157. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.26699804575607171816614970960, −11.03993115865003109388489607335, −10.02465717402818566662076230751, −9.353069944061288098429067035751, −7.34069515056861679350789327389, −6.87575509592591793593268791881, −5.33907914277015605824185692554, −4.41644481697190595592499443806, −3.16845604719373769071179804347, −1.31426435843904977731556452423, 2.41000552652544195334982498373, 3.38748212125772045644349257509, 4.98086458199798684571073047154, 6.28680939640170531926615174835, 6.55249898519486356169793254179, 8.242872620418483026840635081627, 9.253860396596786854840107852982, 10.58932499086138334972378225569, 11.54675081485740784281905938311, 12.35569841600834169468931793849

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