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Properties

Label 2-1152-9.4-c1-0-40
Degree $2$
Conductor $1152$
Sign $0.287 + 0.957i$
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  + (1.72 − 0.101i)3-s + (1.24 − 2.15i)5-s + (−0.909 − 1.57i)7-s + (2.97 − 0.350i)9-s + (0.598 + 1.03i)11-s + (2.83 − 4.90i)13-s + (1.93 − 3.84i)15-s − 5.30·17-s − 4.55·19-s + (−1.73 − 2.63i)21-s + (2.01 − 3.48i)23-s + (−0.589 − 1.02i)25-s + (5.11 − 0.909i)27-s + (3.01 + 5.22i)29-s + (−2.81 + 4.87i)31-s + ⋯
L(s)  = 1  + (0.998 − 0.0585i)3-s + (0.555 − 0.962i)5-s + (−0.343 − 0.595i)7-s + (0.993 − 0.116i)9-s + (0.180 + 0.312i)11-s + (0.785 − 1.36i)13-s + (0.498 − 0.993i)15-s − 1.28·17-s − 1.04·19-s + (−0.377 − 0.574i)21-s + (0.419 − 0.727i)23-s + (−0.117 − 0.204i)25-s + (0.984 − 0.174i)27-s + (0.559 + 0.969i)29-s + (−0.505 + 0.876i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.287 + 0.957i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.448746400\)
\(L(\frac12)\) \(\approx\) \(2.448746400\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.101i)T \)
good5 \( 1 + (-1.24 + 2.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.909 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.598 - 1.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.83 + 4.90i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.30T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 + (-2.01 + 3.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.81 - 4.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 + (-4.57 + 7.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.99 - 6.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.39 + 2.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.54T + 53T^{2} \)
59 \( 1 + (1.85 - 3.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.01 + 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.91 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (4.36 + 7.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.89 - 15.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.455T + 89T^{2} \)
97 \( 1 + (1.01 + 1.76i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.384461110010635218612368576840, −8.760944013464050266329364921166, −8.319062265417109715397735326444, −7.15026745867557615954321779268, −6.44775694471444747921606083093, −5.20943149891760152299756931190, −4.32737491225209375262431404854, −3.38493094019890685810807134977, −2.18305161900686745946492034735, −0.979769174526849387942340839343, 1.92373481928578507740675153995, 2.56901105100359357280290036578, 3.68353810877375608896593984105, 4.52679424795317804898355595286, 6.19529984920133053052208396850, 6.47398925041274031571898470650, 7.45730820270459084200281871733, 8.609114462017368875696376868384, 9.090233378455998343226469879933, 9.742107814680304928928881010345

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