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Properties

Label 2-1152-72.11-c1-0-31
Degree $2$
Conductor $1152$
Sign $0.00922 + 0.999i$
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  + (−0.158 − 1.72i)3-s + (−2.94 + 0.548i)9-s + (4.71 + 2.72i)11-s − 8.02i·17-s + 2.51·19-s + (2.5 − 4.33i)25-s + (1.41 + 4.99i)27-s + (3.94 − 8.57i)33-s + (−0.398 + 0.230i)41-s + (6.45 − 11.1i)43-s + (−3.5 − 6.06i)49-s + (−13.8 + 1.27i)51-s + (−0.398 − 4.33i)57-s + (−8.00 + 4.62i)59-s + (−3.94 − 6.82i)67-s + ⋯
L(s)  = 1  + (−0.0917 − 0.995i)3-s + (−0.983 + 0.182i)9-s + (1.42 + 0.821i)11-s − 1.94i·17-s + 0.575·19-s + (0.5 − 0.866i)25-s + (0.272 + 0.962i)27-s + (0.687 − 1.49i)33-s + (−0.0623 + 0.0359i)41-s + (0.983 − 1.70i)43-s + (−0.5 − 0.866i)49-s + (−1.93 + 0.178i)51-s + (−0.0528 − 0.573i)57-s + (−1.04 + 0.601i)59-s + (−0.481 − 0.833i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.00922 + 0.999i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.00922 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.569710044\)
\(L(\frac12)\) \(\approx\) \(1.569710044\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.158 + 1.72i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.71 - 2.72i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 8.02iT - 17T^{2} \)
19 \( 1 - 2.51T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (0.398 - 0.230i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.45 + 11.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (8.00 - 4.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.94 + 6.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.5 - 9i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (-9.84 + 17.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.348543425189333424178638959238, −8.908877753771020183085476343785, −7.71035286515519312314333069945, −7.06906576892572713991559138231, −6.49895530159600154730911855063, −5.40478287826168305657043277002, −4.45790984099246032811610861511, −3.13229262558257951295816047567, −2.03570364739146042578870178442, −0.78323347839064527267252648058, 1.36035710143235185049055273097, 3.13913852412569725464396858163, 3.83869102940357732242686969519, 4.69842188751858530968350203134, 5.93592144480139243301337136652, 6.28570945145931119887868551936, 7.67348214516689084825022659408, 8.657522311618868622809888928328, 9.141168579412796982199238064862, 9.964857588504347335240455756198

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