Properties

Label 2-2592-36.23-c1-0-22
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  + (1.12 + 0.650i)5-s + (−3.04 + 1.75i)7-s + (0.909 + 1.57i)11-s + (2.62 − 4.54i)13-s − 3.03i·17-s + 3.05i·19-s + (2.61 − 4.52i)23-s + (−1.65 − 2.86i)25-s + (3.24 − 1.87i)29-s + (3.77 + 2.17i)31-s − 4.57·35-s + 1.48·37-s + (−7.32 − 4.22i)41-s + (5.57 − 3.22i)43-s + (6.34 + 10.9i)47-s + ⋯
L(s)  = 1  + (0.504 + 0.291i)5-s + (−1.15 + 0.664i)7-s + (0.274 + 0.475i)11-s + (0.727 − 1.26i)13-s − 0.737i·17-s + 0.701i·19-s + (0.544 − 0.942i)23-s + (−0.330 − 0.572i)25-s + (0.603 − 0.348i)29-s + (0.677 + 0.391i)31-s − 0.773·35-s + 0.244·37-s + (−1.14 − 0.660i)41-s + (0.850 − 0.491i)43-s + (0.926 + 1.60i)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

\(L(1)\) \(\approx\) \(1.798617203\)
\(L(\frac12)\) \(\approx\) \(1.798617203\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 0.650i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.04 - 1.75i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.909 - 1.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.62 + 4.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.03iT - 17T^{2} \)
19 \( 1 - 3.05iT - 19T^{2} \)
23 \( 1 + (-2.61 + 4.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.24 + 1.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.77 - 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.48T + 37T^{2} \)
41 \( 1 + (7.32 + 4.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.57 + 3.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.34 - 10.9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 + (-0.342 + 0.594i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.07 - 3.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.9 - 6.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.50T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (-1.80 + 1.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.58 + 14.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 + (-4.39 - 7.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.961952422698758931627770078031, −8.243629113665205377822789289171, −7.30011577778479531330385812424, −6.36664843605992631801628796509, −6.02993832178097099357164161177, −5.13339869771533910199079696176, −4.00650055437392650224939722922, −2.97719996484185799407421508315, −2.43897014659484611305611413660, −0.841149957307762588212525254426, 0.869826517058519335205367793583, 1.98884045296079542520176031059, 3.35515367127351772838854613881, 3.86882635933181551751126389571, 4.94168787912381993685198008620, 5.92633855568762926024608640788, 6.60898085871308893570841259673, 7.05346700410772507530059799553, 8.271872883771580343816667529017, 8.942467222143808265592404375380

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