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Properties

Label 2-2592-648.283-c0-0-0
Degree $2$
Conductor $2592$
Sign $-0.981 + 0.192i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.286 − 0.957i)3-s + (−0.835 − 0.549i)9-s + (−1.36 − 1.44i)11-s + (−1.67 + 0.611i)17-s + (−1.28 − 0.469i)19-s + (0.893 + 0.448i)25-s + (−0.766 + 0.642i)27-s + (−1.77 + 0.891i)33-s + (0.0333 − 0.572i)41-s + (−0.0333 + 0.111i)43-s + (−0.286 − 0.957i)49-s + (0.103 + 1.78i)51-s + (−0.819 + 1.10i)57-s + (1.33 − 1.41i)59-s + (0.997 + 0.656i)67-s + ⋯
L(s)  = 1  + (0.286 − 0.957i)3-s + (−0.835 − 0.549i)9-s + (−1.36 − 1.44i)11-s + (−1.67 + 0.611i)17-s + (−1.28 − 0.469i)19-s + (0.893 + 0.448i)25-s + (−0.766 + 0.642i)27-s + (−1.77 + 0.891i)33-s + (0.0333 − 0.572i)41-s + (−0.0333 + 0.111i)43-s + (−0.286 − 0.957i)49-s + (0.103 + 1.78i)51-s + (−0.819 + 1.10i)57-s + (1.33 − 1.41i)59-s + (0.997 + 0.656i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-0.981 + 0.192i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (1903, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ -0.981 + 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6654417455\)
\(L(\frac12)\) \(\approx\) \(0.6654417455\)
\(L(1)\) 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 \( 1 + (-0.286 + 0.957i)T \)
good5 \( 1 + (-0.893 - 0.448i)T^{2} \)
7 \( 1 + (0.286 + 0.957i)T^{2} \)
11 \( 1 + (1.36 + 1.44i)T + (-0.0581 + 0.998i)T^{2} \)
13 \( 1 + (-0.597 + 0.802i)T^{2} \)
17 \( 1 + (1.67 - 0.611i)T + (0.766 - 0.642i)T^{2} \)
19 \( 1 + (1.28 + 0.469i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (0.286 - 0.957i)T^{2} \)
29 \( 1 + (-0.396 + 0.918i)T^{2} \)
31 \( 1 + (0.686 + 0.727i)T^{2} \)
37 \( 1 + (-0.173 - 0.984i)T^{2} \)
41 \( 1 + (-0.0333 + 0.572i)T + (-0.993 - 0.116i)T^{2} \)
43 \( 1 + (0.0333 - 0.111i)T + (-0.835 - 0.549i)T^{2} \)
47 \( 1 + (0.686 - 0.727i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.33 + 1.41i)T + (-0.0581 - 0.998i)T^{2} \)
61 \( 1 + (-0.973 - 0.230i)T^{2} \)
67 \( 1 + (-0.997 - 0.656i)T + (0.396 + 0.918i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.290 + 1.64i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.993 - 0.116i)T^{2} \)
83 \( 1 + (-0.0890 - 1.52i)T + (-0.993 + 0.116i)T^{2} \)
89 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.770 + 0.182i)T + (0.893 - 0.448i)T^{2} \)
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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

−8.456502235861317627141385019072, −8.227471427106510737297216003550, −7.09272518182894791442146316941, −6.51311145656982924020768147561, −5.76704217276353562975156847478, −4.88349262873146556529206752224, −3.68001740016595313473626442643, −2.72520498036489660027031426883, −2.00502966388633823089728433316, −0.37448258406067804777208162183, 2.22341723318960544987635510527, 2.67477975095083840688223239387, 4.08173776369170783012353275553, 4.63666790031137925346671219449, 5.23750815480177536103532664959, 6.37425879776917126907266557256, 7.18529358756348378720942944911, 8.074895290865605459544556826377, 8.708210861320219354356986109897, 9.448082958391236907853710241511

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