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Properties

Label 2-2592-9.2-c0-0-1
Degree $2$
Conductor $2592$
Sign $0.819 - 0.573i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.448 + 0.258i)5-s + (0.866 + 1.5i)13-s − 1.93i·17-s + (−0.366 + 0.633i)25-s + (1.67 + 0.965i)29-s + 37-s + (−1.22 + 0.707i)41-s + (0.5 + 0.866i)49-s + 1.41i·53-s + (0.5 − 0.866i)61-s + (−0.776 − 0.448i)65-s + 1.73·73-s + (0.499 + 0.866i)85-s − 0.517i·89-s + (1.22 + 0.707i)101-s + ⋯
L(s)  = 1  + (−0.448 + 0.258i)5-s + (0.866 + 1.5i)13-s − 1.93i·17-s + (−0.366 + 0.633i)25-s + (1.67 + 0.965i)29-s + 37-s + (−1.22 + 0.707i)41-s + (0.5 + 0.866i)49-s + 1.41i·53-s + (0.5 − 0.866i)61-s + (−0.776 − 0.448i)65-s + 1.73·73-s + (0.499 + 0.866i)85-s − 0.517i·89-s + (1.22 + 0.707i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.819 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143280448\)
\(L(\frac12)\) \(\approx\) \(1.143280448\)
\(L(1)\) 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 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.93iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + 0.517iT - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.178551276408931133396533754673, −8.432026215038423314745315404543, −7.52269585422128940768834145896, −6.86236465547285142220376422458, −6.26266300249281401256014736983, −5.07694957747491610508312858302, −4.42710812412909697045054044655, −3.45775083816836195572934637936, −2.58662797013332952472413555942, −1.24501963630514387447146940503, 0.899449167254123472704667751113, 2.27141667596741325084233854029, 3.48311087840086698320941543668, 4.05436710391505562926984871748, 5.10628155876947186140498831197, 5.99138609659569571424169917803, 6.51917410012476296747340190804, 7.74002305394932938022479844726, 8.372990089583820306271713546896, 8.556827471294025860295576868293

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