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Properties

Label 2-2592-9.4-c1-0-42
Degree $2$
Conductor $2592$
Sign $-0.939 - 0.342i$
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  + (−0.5 + 0.866i)5-s + (−1.5 − 2.59i)7-s + (−1.5 − 2.59i)11-s − 4·17-s + 6·19-s + (3 − 5.19i)23-s + (2 + 3.46i)25-s + (−1 − 1.73i)29-s + (−4.5 + 7.79i)31-s + 3·35-s − 2·37-s + (−5 + 8.66i)41-s + (−3 − 5.19i)43-s + (3 + 5.19i)47-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.566 − 0.981i)7-s + (−0.452 − 0.783i)11-s − 0.970·17-s + 1.37·19-s + (0.625 − 1.08i)23-s + (0.400 + 0.692i)25-s + (−0.185 − 0.321i)29-s + (−0.808 + 1.39i)31-s + 0.507·35-s − 0.328·37-s + (−0.780 + 1.35i)41-s + (−0.457 − 0.792i)43-s + (0.437 + 0.757i)47-s + (−0.142 + 0.247i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(20.6972\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2592,\ (\ :1/2),\ -0.939 - 0.342i)\)

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 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.489153319797829438787568010369, −7.52477407668060048299298524339, −6.99198308404082401872359759489, −6.32269933801688119986109175117, −5.27480573604409214460224029556, −4.46568371590275876782488462577, −3.34578572465081479119223222897, −2.96159984522905296786200894002, −1.31603265489514064208339017567, 0, 1.70696821540476642575069897799, 2.70973190262987263803706312392, 3.60441259534869265298399978246, 4.74753163182569142980445251428, 5.34188048773032411748731885712, 6.16890043952399395379181529634, 7.11693158309662915845982184525, 7.72259152564952967517089289230, 8.685067879348353218316216215978

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