Properties

Label 2-1152-12.11-c1-0-6
Degree $2$
Conductor $1152$
Sign $0.816 - 0.577i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + 0.585i·5-s − 0.828i·7-s − 2.82·11-s + 2.82·13-s + 2.58i·17-s + 5.65i·19-s + 6.82·23-s + 4.65·25-s + 3.41i·29-s − 8.82i·31-s + 0.485·35-s + 7.65·37-s + 5.41i·41-s + 1.65i·43-s − 4.48·47-s + ⋯
L(s)  = 1  + 0.261i·5-s − 0.313i·7-s − 0.852·11-s + 0.784·13-s + 0.627i·17-s + 1.29i·19-s + 1.42·23-s + 0.931·25-s + 0.634i·29-s − 1.58i·31-s + 0.0820·35-s + 1.25·37-s + 0.845i·41-s + 0.252i·43-s − 0.654·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586923297\)
\(L(\frac12)\) \(\approx\) \(1.586923297\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585iT - 5T^{2} \)
7 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 2.58iT - 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 - 6.82T + 23T^{2} \)
29 \( 1 - 3.41iT - 29T^{2} \)
31 \( 1 + 8.82iT - 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 5.41iT - 41T^{2} \)
43 \( 1 - 1.65iT - 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 - 9.07iT - 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 - 3.65T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 3.75iT - 89T^{2} \)
97 \( 1 - 2.34T + 97T^{2} \)
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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

−9.962799567892150161201490562542, −9.057793105354217262396810583219, −8.126275158912990645243560800994, −7.53774696270985509759176555930, −6.46277004407426450396466057182, −5.74917625443140913397069840128, −4.67458696876862102829131237348, −3.65536204286048010856906490920, −2.67244674310645454888764126950, −1.19972866375679529908112501987, 0.830953813554536544138076068711, 2.46025955931186664150547554228, 3.34230647187902475315749829755, 4.80047392369487570334180443794, 5.22603830306411940939075843950, 6.45459819917204749709549008344, 7.17995062855596276371499875977, 8.203220991943803733349745743876, 8.924756478024221044963727304835, 9.534351442215529866530562301275

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