Properties

Label 2-2592-36.23-c1-0-38
Degree $2$
Conductor $2592$
Sign $-0.906 + 0.422i$
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  + (0.275 + 0.158i)5-s + (−1.25 + 0.724i)7-s + (−0.548 − 0.949i)11-s + (−1.44 + 2.51i)13-s + 3.46i·17-s − 4.89i·19-s + (1.41 − 2.44i)23-s + (−2.44 − 4.24i)25-s + (−7.89 + 4.56i)29-s + (6.45 + 3.72i)31-s − 0.460·35-s + 4.89·37-s + (−8.44 − 4.87i)41-s + (−5.97 + 3.44i)43-s + (−4.56 − 7.89i)47-s + ⋯
L(s)  = 1  + (0.123 + 0.0710i)5-s + (−0.474 + 0.273i)7-s + (−0.165 − 0.286i)11-s + (−0.402 + 0.696i)13-s + 0.840i·17-s − 1.12i·19-s + (0.294 − 0.510i)23-s + (−0.489 − 0.848i)25-s + (−1.46 + 0.846i)29-s + (1.15 + 0.668i)31-s − 0.0778·35-s + 0.805·37-s + (−1.31 − 0.761i)41-s + (−0.911 + 0.526i)43-s + (−0.665 − 1.15i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.906 + 0.422i$
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: \((2,\ 2592,\ (\ :1/2),\ -0.906 + 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2090166099\)
\(L(\frac12)\) \(\approx\) \(0.2090166099\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.275 - 0.158i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.25 - 0.724i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.548 + 0.949i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.44 - 2.51i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 + (-1.41 + 2.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.89 - 4.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.45 - 3.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + (8.44 + 4.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.97 - 3.44i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.56 + 7.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.41iT - 53T^{2} \)
59 \( 1 + (-4.56 + 7.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.41 + 2.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.56T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + (-1.73 + i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.93 + 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.617592620480385572147814405790, −7.88158920378172006837965059740, −6.73559036795064970655659073695, −6.50431548913372609941660121997, −5.38691921419832954116152532473, −4.65637292415554858248601357495, −3.62838229266459428246985932833, −2.72729507804005166323765089301, −1.72988423699714022868361600197, −0.06602528640683071258317304624, 1.43247724401745467803188392050, 2.66735245335462995189307142728, 3.51383258288459783778970509195, 4.45832995950294297180226915411, 5.42481672753802500275518816003, 6.01697351589339936146500884146, 7.04675393084562095798914634633, 7.66804725637665733853024415962, 8.309510445333311531400021693294, 9.530943417527224239769986057113

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