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Properties

Label 2-1152-72.61-c1-0-27
Degree $2$
Conductor $1152$
Sign $-0.422 + 0.906i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)3-s + (−2 + 3.46i)7-s + (−1.5 − 2.59i)9-s + (−0.866 − 0.5i)11-s + (3.46 − 2i)13-s − 7·17-s + 5i·19-s + (−3.46 − 6i)21-s + (−2 − 3.46i)23-s + (−2.5 + 4.33i)25-s + 5.19·27-s + (3.46 + 2i)29-s + (−4 − 6.92i)31-s + (1.5 − 0.866i)33-s − 8i·37-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.755 + 1.30i)7-s + (−0.5 − 0.866i)9-s + (−0.261 − 0.150i)11-s + (0.960 − 0.554i)13-s − 1.69·17-s + 1.14i·19-s + (−0.755 − 1.30i)21-s + (−0.417 − 0.722i)23-s + (−0.5 + 0.866i)25-s + 1.00·27-s + (0.643 + 0.371i)29-s + (−0.718 − 1.24i)31-s + (0.261 − 0.150i)33-s − 1.31i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.422 + 0.906i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
good5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.46 + 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.46 - 2i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + (7.79 - 4.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.8 + 8i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.461342698668455602767829889671, −8.898487512258585555784143874386, −8.198703976599581314268013076944, −6.73559535660070100796950963029, −5.87032629960682260605032151483, −5.55833982843767662372845153475, −4.22785720497696196667211052070, −3.39659158085874796893480546075, −2.24266118522025334928963041377, 0, 1.37024506425538815992285015538, 2.72277130523598161042220954684, 4.03456380393944183725638612500, 4.86371209358299049337210215971, 6.28551888208523037909502203752, 6.63800177647304519237626200782, 7.36190361563683479198312413726, 8.323580349513391909798481000595, 9.189533806880480795637354018913

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