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Properties

Label 2-2592-36.23-c1-0-0
Degree $2$
Conductor $2592$
Sign $-0.819 - 0.573i$
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  + (−0.896 − 0.517i)5-s + (−0.232 + 0.133i)7-s + (−1.93 − 3.34i)11-s + (1.23 − 2.13i)13-s + 6.69i·17-s − 1.73i·19-s + (−2.96 + 5.13i)23-s + (−1.96 − 3.40i)25-s + (1.79 − 1.03i)29-s + (−0.464 − 0.267i)31-s + 0.277·35-s − 6.46·37-s + (−1.79 − 1.03i)41-s + (6.46 − 3.73i)43-s + (−4.76 − 8.24i)47-s + ⋯
L(s)  = 1  + (−0.400 − 0.231i)5-s + (−0.0877 + 0.0506i)7-s + (−0.582 − 1.00i)11-s + (0.341 − 0.591i)13-s + 1.62i·17-s − 0.397i·19-s + (−0.618 + 1.07i)23-s + (−0.392 − 0.680i)25-s + (0.332 − 0.192i)29-s + (−0.0833 − 0.0481i)31-s + 0.0468·35-s − 1.06·37-s + (−0.280 − 0.161i)41-s + (0.985 − 0.569i)43-s + (−0.694 − 1.20i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(20.6972\)
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: \((2,\ 2592,\ (\ :1/2),\ -0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2248858198\)
\(L(\frac12)\) \(\approx\) \(0.2248858198\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.896 + 0.517i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.232 - 0.133i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.93 + 3.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.23 + 2.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + (2.96 - 5.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.79 + 1.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.464 + 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 + (1.79 + 1.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.46 + 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.76 + 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + (3.72 - 6.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.69 - 8.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.42 - 4.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 9.92T + 73T^{2} \)
79 \( 1 + (13.1 - 7.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.86 + 6.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.69iT - 89T^{2} \)
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

−8.977285280524731535839385437249, −8.320965472589542683174753229010, −7.921710614500767064927247374779, −6.92586978328902063073061175879, −5.83365581368452361479339981177, −5.59871100078018125962233882421, −4.26567838473579882844073009589, −3.62290105400001715203065017483, −2.65064015558349915867543507966, −1.32282276629635955326499121004, 0.07439196177467973919082045760, 1.75117236864064012898646285827, 2.75653642724655033513766547407, 3.72848586742577360610705094838, 4.66825567690974278107704792167, 5.27610915263163003151818633145, 6.49873734220229781898301525418, 7.01670668190056922782639954055, 7.79545799890091725430516447522, 8.461419129765781368780521737229

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