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Properties

Label 2-6e3-72.29-c2-0-20
Degree $2$
Conductor $216$
Sign $-0.343 + 0.938i$
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  + (1.55 − 1.25i)2-s + (0.842 − 3.91i)4-s + (−0.661 − 1.14i)5-s + (4.89 − 8.47i)7-s + (−3.60 − 7.14i)8-s + (−2.47 − 0.952i)10-s + (−6.81 + 11.8i)11-s + (1.13 − 0.654i)13-s + (−3.03 − 19.3i)14-s + (−14.5 − 6.58i)16-s − 0.636i·17-s − 22.9i·19-s + (−5.04 + 1.62i)20-s + (4.22 + 26.9i)22-s + (22.3 − 12.9i)23-s + ⋯
L(s)  = 1  + (0.778 − 0.628i)2-s + (0.210 − 0.977i)4-s + (−0.132 − 0.229i)5-s + (0.699 − 1.21i)7-s + (−0.450 − 0.892i)8-s + (−0.247 − 0.0952i)10-s + (−0.619 + 1.07i)11-s + (0.0872 − 0.0503i)13-s + (−0.216 − 1.38i)14-s + (−0.911 − 0.411i)16-s − 0.0374i·17-s − 1.20i·19-s + (−0.252 + 0.0811i)20-s + (0.192 + 1.22i)22-s + (0.972 − 0.561i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.938i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.343 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.343 + 0.938i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.343 + 0.938i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.30481 - 1.86756i\)
\(L(\frac12)\) \(\approx\) \(1.30481 - 1.86756i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.55 + 1.25i)T \)
3 \( 1 \)
good5 \( 1 + (0.661 + 1.14i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.89 + 8.47i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (6.81 - 11.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.13 + 0.654i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 0.636iT - 289T^{2} \)
19 \( 1 + 22.9iT - 361T^{2} \)
23 \( 1 + (-22.3 + 12.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (6.64 - 11.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-18.7 - 32.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 51.3iT - 1.36e3T^{2} \)
41 \( 1 + (31.8 - 18.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-56.7 - 32.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-75.9 - 43.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 9.23T + 2.80e3T^{2} \)
59 \( 1 + (-14.5 - 25.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-7.53 - 4.35i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (24.3 - 14.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 83.8iT - 5.04e3T^{2} \)
73 \( 1 + 88.8T + 5.32e3T^{2} \)
79 \( 1 + (-22.0 + 38.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-64.9 + 112. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 23.0iT - 7.92e3T^{2} \)
97 \( 1 + (-1.36 + 2.36i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.86586377248304966624836499048, −10.76367631502263681751587616855, −10.33784301052088106326942686017, −9.008618502699808896627383248173, −7.55095692025542680755307331762, −6.64600773696778717746750389428, −4.86272423118910144492869059108, −4.49655201198496664049860171256, −2.79823271790099406768158070857, −1.08542952056873030870367155079, 2.45537152634157809099252971186, 3.76917234967110419121276755448, 5.38715151336511312861676881661, 5.81700838868926323447240338538, 7.33464579492072655184735811108, 8.268538278008904391953237744239, 9.053079555558960505462592111750, 10.82766272671269813533545391680, 11.59541458288853607384592892841, 12.44302648659325866077162383735

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