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Properties

Label 2-1152-9.4-c1-0-26
Degree $2$
Conductor $1152$
Sign $0.684 + 0.729i$
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  + (−0.494 − 1.66i)3-s + (0.268 − 0.464i)5-s + (2.35 + 4.07i)7-s + (−2.51 + 1.64i)9-s + (−2.59 − 4.50i)11-s + (−0.778 + 1.34i)13-s + (−0.903 − 0.215i)15-s + 0.695·17-s + 5.80·19-s + (5.59 − 5.91i)21-s + (4.42 − 7.66i)23-s + (2.35 + 4.08i)25-s + (3.96 + 3.35i)27-s + (1.92 + 3.32i)29-s + (2.77 − 4.79i)31-s + ⋯
L(s)  = 1  + (−0.285 − 0.958i)3-s + (0.119 − 0.207i)5-s + (0.888 + 1.53i)7-s + (−0.837 + 0.546i)9-s + (−0.783 − 1.35i)11-s + (−0.215 + 0.373i)13-s + (−0.233 − 0.0556i)15-s + 0.168·17-s + 1.33·19-s + (1.22 − 1.29i)21-s + (0.923 − 1.59i)23-s + (0.471 + 0.816i)25-s + (0.763 + 0.646i)27-s + (0.356 + 0.618i)29-s + (0.497 − 0.861i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.684 + 0.729i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.597792411\)
\(L(\frac12)\) \(\approx\) \(1.597792411\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.494 + 1.66i)T \)
good5 \( 1 + (-0.268 + 0.464i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.35 - 4.07i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 + 4.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.778 - 1.34i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.695T + 17T^{2} \)
19 \( 1 - 5.80T + 19T^{2} \)
23 \( 1 + (-4.42 + 7.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.92 - 3.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.77 + 4.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + (-1.01 + 1.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.71 + 6.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.186 + 0.322i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.30T + 53T^{2} \)
59 \( 1 + (2.57 - 4.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.921 + 1.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.79 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 4.40T + 73T^{2} \)
79 \( 1 + (-3.32 - 5.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.28 - 9.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.30T + 89T^{2} \)
97 \( 1 + (7.81 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.456695311353804232284082737134, −8.543627372278999686070469853659, −8.281476981506927909245937540439, −7.24641148840109096321628024072, −6.21115041174780760911203458648, −5.39347668458116727424859013797, −5.02161578435267839895831510857, −3.02268724977056940542222946968, −2.29805316092972625743354871523, −0.935520871809729528226655410648, 1.10371555436788300006204605223, 2.82991745296736925162951298806, 3.91006017028093909985945892258, 4.87083900652564518554511785174, 5.20245157472469102941514961806, 6.67932974827991555917352359870, 7.53491014282693008825012033922, 8.058420955068326507008750254129, 9.469383896485171155022942509871, 10.01488383663615333696704675989

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