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Properties

Label 2-2592-36.11-c1-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67 − 0.965i)5-s + (−0.633 − 0.366i)7-s + (−2.63 + 4.57i)11-s + (−2.23 − 3.86i)13-s − 0.896i·17-s + 1.26i·19-s + (−0.707 − 1.22i)23-s + (−0.633 + 1.09i)25-s + (−4.69 − 2.70i)29-s + (−6.46 + 3.73i)31-s − 1.41·35-s − 7.73·37-s + (−0.328 + 0.189i)41-s + (7.56 + 4.36i)43-s + (2.31 − 4.00i)47-s + ⋯
L(s)  = 1  + (0.748 − 0.431i)5-s + (−0.239 − 0.138i)7-s + (−0.795 + 1.37i)11-s + (−0.619 − 1.07i)13-s − 0.217i·17-s + 0.290i·19-s + (−0.147 − 0.255i)23-s + (−0.126 + 0.219i)25-s + (−0.871 − 0.502i)29-s + (−1.16 + 0.670i)31-s − 0.239·35-s − 1.27·37-s + (−0.0512 + 0.0295i)41-s + (1.15 + 0.665i)43-s + (0.337 − 0.583i)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: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.965i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.633 + 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.63 - 4.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.23 + 3.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.896iT - 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.69 + 2.70i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.73T + 37T^{2} \)
41 \( 1 + (0.328 - 0.189i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.56 - 4.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.31 + 4.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.44iT - 53T^{2} \)
59 \( 1 + (5.41 + 9.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.598 + 1.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.3 - 6.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 + (14.3 + 8.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.27 - 9.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 + (-3.19 + 5.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.528845846580699460231769333047, −7.50215939901647362675114265415, −7.22402055818852173299999586885, −5.97883477918906309222378804847, −5.31762207916291406726784878059, −4.73626971166252665145012317524, −3.56616864503069982950573393132, −2.47551565278196621880362134921, −1.65638238273228411627424710877, 0, 1.76290315061534597238104821109, 2.64600825892267336006168006894, 3.51572002415159158679418570045, 4.58796841447680306114958672245, 5.75906658501254577067014250163, 5.91989239523966112406149197416, 7.04985495116325049782409241356, 7.61789382024341859223143345213, 8.762430851557246865979118327576

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