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Properties

Label 2-864-24.11-c1-0-3
Degree $2$
Conductor $864$
Sign $-0.280 - 0.959i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.505·5-s + 3.42i·7-s + 3.31i·11-s + 2.55i·13-s − 5.04i·17-s − 4.74·19-s − 6.44·23-s − 4.74·25-s + 5.43·29-s + 5.97i·31-s + 1.73i·35-s + 11.1i·37-s + 1.87i·41-s + 4·43-s + 10.8·47-s + ⋯
L(s)  = 1  + 0.226·5-s + 1.29i·7-s + 1.00i·11-s + 0.707i·13-s − 1.22i·17-s − 1.08·19-s − 1.34·23-s − 0.948·25-s + 1.00·29-s + 1.07i·31-s + 0.292i·35-s + 1.83i·37-s + 0.293i·41-s + 0.609·43-s + 1.58·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.280 - 0.959i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.280 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738765 + 0.985144i\)
\(L(\frac12)\) \(\approx\) \(0.738765 + 0.985144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 0.505T + 5T^{2} \)
7 \( 1 - 3.42iT - 7T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 - 2.55iT - 13T^{2} \)
17 \( 1 + 5.04iT - 17T^{2} \)
19 \( 1 + 4.74T + 19T^{2} \)
23 \( 1 + 6.44T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 - 5.97iT - 31T^{2} \)
37 \( 1 - 11.1iT - 37T^{2} \)
41 \( 1 - 1.87iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 5.93T + 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 - 5.10iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 7.74T + 73T^{2} \)
79 \( 1 - 4.30iT - 79T^{2} \)
83 \( 1 + 3.61iT - 83T^{2} \)
89 \( 1 - 17.0iT - 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.19704720544871466773545078111, −9.574014186519946199219374072243, −8.808546626057689788333096364183, −7.989030605245361071258136936261, −6.84898838899732937431754188992, −6.13105162813058636775103692253, −5.09038240185952142444288800844, −4.26336714120315615373162669209, −2.70652671464320318312253836531, −1.89294673172173153945813497113, 0.57522832837880327525537985606, 2.16298109142292269839949396496, 3.71252888916599222679239128433, 4.22591597986770543772465928908, 5.79733254312686691672209068137, 6.24256507888734994023347359501, 7.54250586709214242709118302309, 8.097741661778216064832175937548, 9.030766261544192369248911051887, 10.29950523232251477978564970960

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