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Properties

Label 2-2592-9.4-c1-0-41
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  + (1 − 1.73i)5-s + (−2 − 3.46i)7-s + (−2 − 3.46i)11-s + (1 − 1.73i)13-s + 6·17-s + 4·19-s + (0.500 + 0.866i)25-s + (1 + 1.73i)29-s + (2 − 3.46i)31-s − 7.99·35-s − 2·37-s + (1 − 1.73i)41-s + (2 + 3.46i)43-s + (−4 − 6.92i)47-s + (−4.49 + 7.79i)49-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (−0.755 − 1.30i)7-s + (−0.603 − 1.04i)11-s + (0.277 − 0.480i)13-s + 1.45·17-s + 0.917·19-s + (0.100 + 0.173i)25-s + (0.185 + 0.321i)29-s + (0.359 − 0.622i)31-s − 1.35·35-s − 0.328·37-s + (0.156 − 0.270i)41-s + (0.304 + 0.528i)43-s + (−0.583 − 1.01i)47-s + (−0.642 + 1.11i)49-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: $\chi_{2592} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.561893445\)
\(L(\frac12)\) \(\approx\) \(1.561893445\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.523196761884638693677920060893, −7.82563842306841597038728593252, −7.18481242833963500625169124498, −6.10979780039144253905857067493, −5.53378037272390217592388190967, −4.71883769099416620556295262357, −3.49839209218241177546766190160, −3.11210321750689923638245189346, −1.33153106367843797436547517728, −0.54623318411897654023694048051, 1.61243274288161144511544500488, 2.75777918642639635223666845750, 3.11594811809659479814376814325, 4.51376439229060954432598363816, 5.51606420923729636508185604890, 6.01006486638016017582975513082, 6.86166440945316610047332621942, 7.55126874611262419808969959611, 8.466770107578634234459457748071, 9.353773918829635163119583115159

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