Properties

Label 2-2592-9.7-c1-0-3
Degree $2$
Conductor $2592$
Sign $-0.984 - 0.173i$
Analytic cond. $20.6972$
Root an. cond. $4.54942$
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  + (1.86 + 3.23i)5-s + (−3.23 − 5.59i)13-s − 5.73·17-s + (−4.46 + 7.73i)25-s + (−5.33 + 9.23i)29-s − 9.39·37-s + (4 + 6.92i)41-s + (3.5 + 6.06i)49-s − 4·53-s + (−7.69 + 13.3i)61-s + (12.0 − 20.8i)65-s − 16.8·73-s + (−10.6 − 18.5i)85-s + 0.660·89-s + (9 − 15.5i)97-s + ⋯
L(s)  = 1  + (0.834 + 1.44i)5-s + (−0.896 − 1.55i)13-s − 1.39·17-s + (−0.892 + 1.54i)25-s + (−0.989 + 1.71i)29-s − 1.54·37-s + (0.624 + 1.08i)41-s + (0.5 + 0.866i)49-s − 0.549·53-s + (−0.985 + 1.70i)61-s + (1.49 − 2.59i)65-s − 1.97·73-s + (−1.16 − 2.00i)85-s + 0.0699·89-s + (0.913 − 1.58i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7415699051\)
\(L(\frac12)\) \(\approx\) \(0.7415699051\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-1.86 - 3.23i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.23 + 5.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.33 - 9.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.39T + 37T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.69 - 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.660T + 89T^{2} \)
97 \( 1 + (-9 + 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.330798775396211208232004732176, −8.546759064947546862414366962203, −7.33826199097136474189966073399, −7.14869041813359522363017799805, −6.12511024314716596194618031746, −5.54799508037633283865008283116, −4.57937427688622491642882071640, −3.22908293305832255919377670398, −2.76953162033645582529117556317, −1.74961031628506591881867834168, 0.21473210720472761115649899956, 1.79904868561959792384250398213, 2.22869822164705421015879387963, 3.98010544142208454154197734933, 4.61027654244040808926046705488, 5.27889313501678102265872241344, 6.17131598576147665210803968424, 6.91140716446833286614659036996, 7.84208562249918207578664955748, 8.875420310525524341848713428285

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