Properties

Label 2-2592-36.11-c1-0-38
Degree $2$
Conductor $2592$
Sign $0.573 + 0.819i$
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  + (3.34 − 1.93i)5-s + (3.23 + 1.86i)7-s + (−0.517 + 0.896i)11-s + (−2.23 − 3.86i)13-s − 1.79i·17-s + 1.73i·19-s + (−4.38 − 7.58i)23-s + (4.96 − 8.59i)25-s + (−6.69 − 3.86i)29-s + (6.46 − 3.73i)31-s + 14.4·35-s + 0.464·37-s + (6.69 − 3.86i)41-s + (−0.464 − 0.267i)43-s + (2.31 − 4.00i)47-s + ⋯
L(s)  = 1  + (1.49 − 0.863i)5-s + (1.22 + 0.705i)7-s + (−0.156 + 0.270i)11-s + (−0.619 − 1.07i)13-s − 0.434i·17-s + 0.397i·19-s + (−0.913 − 1.58i)23-s + (0.992 − 1.71i)25-s + (−1.24 − 0.717i)29-s + (1.16 − 0.670i)31-s + 2.43·35-s + 0.0762·37-s + (1.04 − 0.603i)41-s + (−0.0707 − 0.0408i)43-s + (0.337 − 0.583i)47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.643329783\)
\(L(\frac12)\) \(\approx\) \(2.643329783\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.34 + 1.93i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.23 - 1.86i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.517 - 0.896i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.23 + 3.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.79iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (4.38 + 7.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.69 + 3.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.46 + 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.464T + 37T^{2} \)
41 \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.464 + 0.267i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.31 + 4.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.58iT - 53T^{2} \)
59 \( 1 + (-6.17 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.69 - 9.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.42 - 3.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 3.92T + 73T^{2} \)
79 \( 1 + (-4.16 - 2.40i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.03 - 1.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.79iT - 89T^{2} \)
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

−8.691782035568216035139670856918, −8.199277535184559904661555803054, −7.40145476306966817553463797019, −6.06662455309002932019065203201, −5.66022251215817741554498436354, −4.97602856410734868009978595693, −4.28327376742755474907367135752, −2.42706204143799214884577337473, −2.18881452094541653214582722253, −0.869576036053740533631044052665, 1.51874831619176091060084567020, 2.03984795004479101269251531515, 3.19221474523152185786104046603, 4.32094639465065396709609726596, 5.16130682871462482447516636504, 5.92375833921649657942681802537, 6.70905319697283848695612145293, 7.41386375576227978671333102856, 8.109399332836210039046701333669, 9.279250424186919169332737846423

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