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Properties

Label 2-2592-9.7-c1-0-8
Degree $2$
Conductor $2592$
Sign $0.939 - 0.342i$
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  + (−1.80 − 3.12i)5-s + (−1.80 + 3.12i)7-s + (−0.5 + 0.866i)11-s + (−2 − 3.46i)13-s − 7.21·19-s + (3 + 5.19i)23-s + (−4 + 6.92i)25-s + (3.60 − 6.24i)29-s + (1.80 + 3.12i)31-s + 13·35-s + 10·37-s + (3.60 + 6.24i)41-s + (−3.60 + 6.24i)43-s + (5 − 8.66i)47-s + (−3 − 5.19i)49-s + ⋯
L(s)  = 1  + (−0.806 − 1.39i)5-s + (−0.681 + 1.18i)7-s + (−0.150 + 0.261i)11-s + (−0.554 − 0.960i)13-s − 1.65·19-s + (0.625 + 1.08i)23-s + (−0.800 + 1.38i)25-s + (0.669 − 1.15i)29-s + (0.323 + 0.560i)31-s + 2.19·35-s + 1.64·37-s + (0.563 + 0.975i)41-s + (−0.549 + 0.952i)43-s + (0.729 − 1.26i)47-s + (−0.428 − 0.742i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9588779629\)
\(L(\frac12)\) \(\approx\) \(0.9588779629\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 3.12i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.80 - 3.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.60 + 6.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.80 - 3.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-3.60 - 6.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.60 - 6.24i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.60 + 6.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + (-7.21 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.21T + 89T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.908237986554297108614605820909, −8.152698526501319699806838597961, −7.74557149449545430680109088263, −6.48246267015711000543475393398, −5.75737327333538372591264318711, −4.92620690427994051918874938635, −4.33183883921067075881604374366, −3.17681865538028980820722463136, −2.24978732367439604043979062366, −0.74984779901455562888970875210, 0.47050646583233183624933664408, 2.34252387284092740259132653578, 3.12371203391964638530795861330, 4.08495196972266975303787195601, 4.48904074832209216485787042667, 6.09915705861933772953636236383, 6.82416164606607026026301800648, 7.04110172476069763434420806863, 7.921517968627099822507891114864, 8.776282610068169466172511160833

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