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Properties

Label 2-6e3-72.5-c2-0-11
Degree $2$
Conductor $216$
Sign $0.392 + 0.919i$
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  + (−0.676 + 1.88i)2-s + (−3.08 − 2.54i)4-s + (−4.28 + 7.41i)5-s + (−3.75 − 6.50i)7-s + (6.87 − 4.08i)8-s + (−11.0 − 13.0i)10-s + (−4.74 − 8.22i)11-s + (9.54 + 5.51i)13-s + (14.7 − 2.67i)14-s + (3.04 + 15.7i)16-s − 11.3i·17-s − 18.3i·19-s + (32.0 − 11.9i)20-s + (18.6 − 3.37i)22-s + (−22.8 − 13.1i)23-s + ⋯
L(s)  = 1  + (−0.338 + 0.941i)2-s + (−0.771 − 0.636i)4-s + (−0.856 + 1.48i)5-s + (−0.536 − 0.929i)7-s + (0.859 − 0.510i)8-s + (−1.10 − 1.30i)10-s + (−0.431 − 0.747i)11-s + (0.734 + 0.424i)13-s + (1.05 − 0.190i)14-s + (0.190 + 0.981i)16-s − 0.667i·17-s − 0.967i·19-s + (1.60 − 0.599i)20-s + (0.849 − 0.153i)22-s + (−0.993 − 0.573i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.392 + 0.919i$
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: \((2,\ 216,\ (\ :1),\ 0.392 + 0.919i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269913 - 0.178196i\)
\(L(\frac12)\) \(\approx\) \(0.269913 - 0.178196i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.676 - 1.88i)T \)
3 \( 1 \)
good5 \( 1 + (4.28 - 7.41i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.75 + 6.50i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.74 + 8.22i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.54 - 5.51i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 11.3iT - 289T^{2} \)
19 \( 1 + 18.3iT - 361T^{2} \)
23 \( 1 + (22.8 + 13.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-3.48 - 6.03i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.42 + 11.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 5.89iT - 1.36e3T^{2} \)
41 \( 1 + (32.7 + 18.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (21.1 - 12.2i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-15.8 + 9.17i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 58.4T + 2.80e3T^{2} \)
59 \( 1 + (13.1 - 22.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (56.0 - 32.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (28.4 + 16.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 84.5iT - 5.04e3T^{2} \)
73 \( 1 + 94.1T + 5.32e3T^{2} \)
79 \( 1 + (12.3 + 21.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (57.7 + 100. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-94.1 - 163. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.60826805378763017906918193287, −10.73051923078462356419866736355, −10.08939657849804990900189287037, −8.702899690293877674138682429498, −7.62404194764562405201862185457, −6.91142633736804357071293717749, −6.13837991528231829962382723717, −4.34058781907187026299878499803, −3.21194745092258268908959173812, −0.20560702926969296512009697836, 1.62798055785338626577973900993, 3.46107983073837619687635922038, 4.57197123291804185978882362608, 5.74814482591263945149463130243, 7.85969381286626506783161496582, 8.414561591378393940325033841134, 9.327238521587073310655788064460, 10.24951905429068971087012418562, 11.55344292449274064464708711670, 12.45980120410875540004210139384

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