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Properties

Label 2-2592-36.23-c1-0-17
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  + (−2.35 − 1.35i)5-s + (3.67 − 2.12i)7-s + (−1.09 − 1.90i)11-s + (−1.25 + 2.17i)13-s + 7.93i·17-s + 5.13i·19-s + (−0.868 + 1.50i)23-s + (1.18 + 2.05i)25-s + (−0.230 + 0.133i)29-s + (8.69 + 5.01i)31-s − 11.5·35-s + 9.24·37-s + (2.18 + 1.26i)41-s + (−1.14 + 0.659i)43-s + (0.860 + 1.49i)47-s + ⋯
L(s)  = 1  + (−1.05 − 0.607i)5-s + (1.39 − 0.802i)7-s + (−0.331 − 0.574i)11-s + (−0.348 + 0.604i)13-s + 1.92i·17-s + 1.17i·19-s + (−0.181 + 0.313i)23-s + (0.237 + 0.411i)25-s + (−0.0428 + 0.0247i)29-s + (1.56 + 0.901i)31-s − 1.94·35-s + 1.52·37-s + (0.341 + 0.196i)41-s + (−0.174 + 0.100i)43-s + (0.125 + 0.217i)47-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: $\chi_{2592} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.582668037\)
\(L(\frac12)\) \(\approx\) \(1.582668037\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.35 + 1.35i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.67 + 2.12i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.09 + 1.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.93iT - 17T^{2} \)
19 \( 1 - 5.13iT - 19T^{2} \)
23 \( 1 + (0.868 - 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.230 - 0.133i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.69 - 5.01i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.24T + 37T^{2} \)
41 \( 1 + (-2.18 - 1.26i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.14 - 0.659i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.860 - 1.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.306iT - 53T^{2} \)
59 \( 1 + (-5.83 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.64 + 8.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.59 + 3.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 6.62T + 73T^{2} \)
79 \( 1 + (9.83 - 5.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.41 - 11.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.42iT - 89T^{2} \)
97 \( 1 + (4.12 + 7.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.454531925555805625099577802161, −8.077927572563799546689993981942, −7.83032595029089801545474588546, −6.65441879649144945621337221532, −5.74499564275138783752124401602, −4.68313569318565653622139242540, −4.24442629219639400534302359527, −3.49255937883504919484118681684, −1.88589338728620168306748862628, −0.973191578320899482127155512000, 0.67827127908361515144900980051, 2.46822088859173918294593189363, 2.79090955057513920209299126164, 4.32246443630168142850881761156, 4.80127141421787160170753784014, 5.58326734830437285805309998937, 6.77662700408302807090415249156, 7.62808573068051305900221897220, 7.80480478558157691511176567423, 8.773266286320048117378115455332

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