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Properties

Label 2-2592-72.11-c1-0-4
Degree $2$
Conductor $2592$
Sign $-0.766 - 0.642i$
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  + (−2.44 − 1.41i)11-s + 5.65i·17-s − 2·19-s + (2.5 − 4.33i)25-s + (−9.79 + 5.65i)41-s + (−5 + 8.66i)43-s + (−3.5 − 6.06i)49-s + (−12.2 + 7.07i)59-s + (7 + 12.1i)67-s + 2·73-s + (−2.44 − 1.41i)83-s + 5.65i·89-s + (5 − 8.66i)97-s + 19.7i·107-s + (−9.79 + 5.65i)113-s + ⋯
L(s)  = 1  + (−0.738 − 0.426i)11-s + 1.37i·17-s − 0.458·19-s + (0.5 − 0.866i)25-s + (−1.53 + 0.883i)41-s + (−0.762 + 1.32i)43-s + (−0.5 − 0.866i)49-s + (−1.59 + 0.920i)59-s + (0.855 + 1.48i)67-s + 0.234·73-s + (−0.268 − 0.155i)83-s + 0.599i·89-s + (0.507 − 0.879i)97-s + 1.91i·107-s + (−0.921 + 0.532i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5629297846\)
\(L(\frac12)\) \(\approx\) \(0.5629297846\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (9.79 - 5.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (12.2 - 7.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.44 + 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.047406170280449567976205008038, −8.274403351211712973537488138858, −7.931853190227023061418158335260, −6.71774696459515751622067079581, −6.20289576326019522772281618704, −5.27004094734669202878394711164, −4.45983532116325001945832221258, −3.49620111931684390386587333473, −2.57658953095523455543862439495, −1.42311658132474521531030589240, 0.17890990506555832605726240608, 1.74848264484377755100651238595, 2.77615408748829096392008289601, 3.64836071225260536695772895522, 4.89629211689478944383308980025, 5.19663322315492954939280318191, 6.37785797587297543818014873575, 7.12677903569311901436779281844, 7.73724118266240200983656237431, 8.629927783566263312837545245277

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