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Properties

Label 2-2592-36.23-c1-0-12
Degree $2$
Conductor $2592$
Sign $-0.173 - 0.984i$
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  + (1.67 + 0.965i)5-s + (0.633 − 0.366i)7-s + (2.63 + 4.57i)11-s + (−2.23 + 3.86i)13-s + 0.896i·17-s + 1.26i·19-s + (0.707 − 1.22i)23-s + (−0.633 − 1.09i)25-s + (−4.69 + 2.70i)29-s + (6.46 + 3.73i)31-s + 1.41·35-s − 7.73·37-s + (−0.328 − 0.189i)41-s + (−7.56 + 4.36i)43-s + (−2.31 − 4.00i)47-s + ⋯
L(s)  = 1  + (0.748 + 0.431i)5-s + (0.239 − 0.138i)7-s + (0.795 + 1.37i)11-s + (−0.619 + 1.07i)13-s + 0.217i·17-s + 0.290i·19-s + (0.147 − 0.255i)23-s + (−0.126 − 0.219i)25-s + (−0.871 + 0.502i)29-s + (1.16 + 0.670i)31-s + 0.239·35-s − 1.27·37-s + (−0.0512 − 0.0295i)41-s + (−1.15 + 0.665i)43-s + (−0.337 − 0.583i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851755711\)
\(L(\frac12)\) \(\approx\) \(1.851755711\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 0.965i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.633 + 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.63 - 4.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.23 - 3.86i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.896iT - 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.69 - 2.70i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.46 - 3.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.73T + 37T^{2} \)
41 \( 1 + (0.328 + 0.189i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.56 - 4.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.31 + 4.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.44iT - 53T^{2} \)
59 \( 1 + (-5.41 + 9.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.598 - 1.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 + (-14.3 + 8.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.27 - 9.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 + (-3.19 - 5.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.193079063763799036303708870759, −8.427678260289475896310816795160, −7.33730827587238896296375796647, −6.79879800619295762077612606171, −6.22996390513403425130400734614, −5.03005276955646709003136614646, −4.47176987215882781327933071957, −3.43556494889887153565472478914, −2.14312221560437478468116695043, −1.62067294608796109163379061787, 0.58784539513444943752777020085, 1.74434240069690911075688988717, 2.89386608657412599745947270371, 3.74129090185368473242214746724, 4.94875162855910473691114151274, 5.56202906933954939832983170522, 6.18364210501693227725407726482, 7.13473660733765198956101361953, 8.078911849970623360924626508935, 8.670977035000622369437018536075

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