Properties

Label 2-2592-36.11-c1-0-46
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  + (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

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

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: $\chi_{2592} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424217703\)
\(L(\frac12)\) \(\approx\) \(1.424217703\)
\(L(\frac{3}{2})\) 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

−8.964039085575768450282137849545, −7.64991527721019172643578187125, −7.08624884571755496167970366145, −6.04616171405843982326523812007, −5.56969119082098226436653056596, −4.83049841654072543309876653002, −3.58415818743394112554873202381, −2.78472338743213677801880109541, −1.56711439493954975518367415386, −0.43115137269239682622422149310, 1.78950819188395781485461708774, 2.49103777139732255261172446623, 3.26773249999508193622474168558, 4.47083527278406768476409178661, 5.63582350532599118533710198799, 6.10892475152240024541516995193, 6.77868161869313004129007442586, 7.33137680256421391200382373963, 8.806992429726200180648876341566, 9.345148090242428759223961453738

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