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Properties

Label 2-1152-9.4-c1-0-19
Degree $2$
Conductor $1152$
Sign $0.107 - 0.994i$
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  + (1.71 + 0.243i)3-s + (−1.34 + 2.32i)5-s + (2.48 + 4.30i)7-s + (2.88 + 0.835i)9-s + (−1.26 − 2.19i)11-s + (2.21 − 3.83i)13-s + (−2.86 + 3.65i)15-s − 2.43·17-s + 4.18·19-s + (3.21 + 7.99i)21-s + (−0.570 + 0.988i)23-s + (−1.09 − 1.89i)25-s + (4.73 + 2.13i)27-s + (3.00 + 5.20i)29-s + (−2.65 + 4.59i)31-s + ⋯
L(s)  = 1  + (0.990 + 0.140i)3-s + (−0.599 + 1.03i)5-s + (0.939 + 1.62i)7-s + (0.960 + 0.278i)9-s + (−0.382 − 0.662i)11-s + (0.614 − 1.06i)13-s + (−0.739 + 0.943i)15-s − 0.590·17-s + 0.959·19-s + (0.701 + 1.74i)21-s + (−0.118 + 0.206i)23-s + (−0.218 − 0.378i)25-s + (0.911 + 0.410i)27-s + (0.557 + 0.966i)29-s + (−0.476 + 0.825i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.107 - 0.994i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.107 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.307977460\)
\(L(\frac12)\) \(\approx\) \(2.307977460\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.243i)T \)
good5 \( 1 + (1.34 - 2.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.48 - 4.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.26 + 2.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.21 + 3.83i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.43T + 17T^{2} \)
19 \( 1 - 4.18T + 19T^{2} \)
23 \( 1 + (0.570 - 0.988i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.00 - 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.65 - 4.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.241T + 37T^{2} \)
41 \( 1 + (3.21 - 5.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.57 + 9.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.37 + 4.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 + (-5.40 + 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 7.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.13 - 1.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + (3.14 + 5.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.738 + 1.27i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + (-5.89 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.997896340100041243447079147117, −8.797160359239285492530082304934, −8.470794728828447708155611103170, −7.74733466829964347386536149862, −6.84719903726292561304360219828, −5.64658191163615745145320972821, −4.89072766689338613516112433386, −3.29410815072154492481064454018, −3.05186464640681938007548757389, −1.80419021467623088858921925332, 0.972804908383088517223607528807, 1.94708046151848859554131640247, 3.62184788843428865791370583494, 4.43673910946034153847196959253, 4.73806358879758609061965837320, 6.56452863828157092801488720005, 7.47706456909388674580892688454, 7.915407429491164271206592118729, 8.638567720524405498048792835192, 9.505837074521635118416341371127

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