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Properties

Label 2-2592-36.11-c1-0-18
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: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.174666765\)
\(L(\frac12)\) \(\approx\) \(1.174666765\)
\(L(\frac{3}{2})\) 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

−9.148869067432296700766934335855, −7.88413873328786641688667616501, −7.33600252306211286524455122919, −6.56260173035119439117679003317, −6.06236245000100395881725025179, −4.64987302072018602834408154931, −4.08409885390102857610455922532, −3.21118143527075838199939458457, −2.21820424147524527369122680490, −0.61832175131282115662167844638, 0.71307486065703266584523671089, 2.25579855555686100732712935535, 3.45405755665892784234677348987, 3.72596796925416040428766110959, 5.16932957980223477393938713598, 5.90606989107499556087413283931, 6.34852648799129467616885185267, 7.56121797251696778840274855285, 8.253393055942486484498487760013, 8.696195772760102713161921418756

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