Properties

Label 2-6e3-24.5-c2-0-16
Degree $2$
Conductor $216$
Sign $0.988 - 0.148i$
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  + (−1.68 + 1.08i)2-s + (1.64 − 3.64i)4-s + 8.89·5-s − 3.35·7-s + (1.19 + 7.91i)8-s + (−14.9 + 9.64i)10-s + 8.12·11-s − 22.5i·13-s + (5.63 − 3.63i)14-s + (−10.5 − 12.0i)16-s − 14.4i·17-s + 13.7i·19-s + (14.6 − 32.4i)20-s + (−13.6 + 8.81i)22-s + 29.4i·23-s + ⋯
L(s)  = 1  + (−0.840 + 0.542i)2-s + (0.411 − 0.911i)4-s + 1.77·5-s − 0.479·7-s + (0.148 + 0.988i)8-s + (−1.49 + 0.964i)10-s + 0.738·11-s − 1.73i·13-s + (0.402 − 0.259i)14-s + (−0.661 − 0.750i)16-s − 0.848i·17-s + 0.721i·19-s + (0.731 − 1.62i)20-s + (−0.620 + 0.400i)22-s + 1.28i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.988 - 0.148i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.38155 + 0.103361i\)
\(L(\frac12)\) \(\approx\) \(1.38155 + 0.103361i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.68 - 1.08i)T \)
3 \( 1 \)
good5 \( 1 - 8.89T + 25T^{2} \)
7 \( 1 + 3.35T + 49T^{2} \)
11 \( 1 - 8.12T + 121T^{2} \)
13 \( 1 + 22.5iT - 169T^{2} \)
17 \( 1 + 14.4iT - 289T^{2} \)
19 \( 1 - 13.7iT - 361T^{2} \)
23 \( 1 - 29.4iT - 529T^{2} \)
29 \( 1 - 11.0T + 841T^{2} \)
31 \( 1 - 41.8T + 961T^{2} \)
37 \( 1 - 11.3iT - 1.36e3T^{2} \)
41 \( 1 - 4.33iT - 1.68e3T^{2} \)
43 \( 1 + 54.4iT - 1.84e3T^{2} \)
47 \( 1 - 12.3iT - 2.20e3T^{2} \)
53 \( 1 - 50.1T + 2.80e3T^{2} \)
59 \( 1 + 49.5T + 3.48e3T^{2} \)
61 \( 1 - 67.9iT - 3.72e3T^{2} \)
67 \( 1 + 80.2iT - 4.48e3T^{2} \)
71 \( 1 - 81.9iT - 5.04e3T^{2} \)
73 \( 1 - 11.9T + 5.32e3T^{2} \)
79 \( 1 + 37.2T + 6.24e3T^{2} \)
83 \( 1 + 58.3T + 6.88e3T^{2} \)
89 \( 1 - 92.5iT - 7.92e3T^{2} \)
97 \( 1 + 14.2T + 9.40e3T^{2} \)
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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

−12.07065423627410206972361620814, −10.63899999108113162068604617370, −9.903515185463316954725783676134, −9.400543375963469404992210517688, −8.239931081430355139579386887503, −6.93697006878191468660909306569, −5.96628613467945729262976611942, −5.31544376198299227527552330736, −2.79173161792555532973673127578, −1.22135989553702130014898472700, 1.50414731718866756459025161508, 2.61754347277584465612549158480, 4.40179883723399128908829166168, 6.37668663311296111912094058966, 6.66987107224513586192650867168, 8.577570531238925520854502013424, 9.314971697948735516928088271319, 9.925261399003962944483727289950, 10.86657957469213984891902586128, 11.98100835619579226387140133299

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