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Properties

Label 2-2592-648.475-c0-0-0
Degree $2$
Conductor $2592$
Sign $0.987 - 0.154i$
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  + (−0.597 − 0.802i)3-s + (−0.286 + 0.957i)9-s + (0.0460 − 0.106i)11-s + (0.337 + 1.91i)17-s + (−0.137 + 0.780i)19-s + (0.973 + 0.230i)25-s + (0.939 − 0.342i)27-s + (−0.113 + 0.0268i)33-s + (−0.819 − 0.868i)41-s + (0.819 + 1.10i)43-s + (0.597 − 0.802i)49-s + (1.33 − 1.41i)51-s + (0.707 − 0.355i)57-s + (0.786 + 1.82i)59-s + (0.512 − 1.71i)67-s + ⋯
L(s)  = 1  + (−0.597 − 0.802i)3-s + (−0.286 + 0.957i)9-s + (0.0460 − 0.106i)11-s + (0.337 + 1.91i)17-s + (−0.137 + 0.780i)19-s + (0.973 + 0.230i)25-s + (0.939 − 0.342i)27-s + (−0.113 + 0.0268i)33-s + (−0.819 − 0.868i)41-s + (0.819 + 1.10i)43-s + (0.597 − 0.802i)49-s + (1.33 − 1.41i)51-s + (0.707 − 0.355i)57-s + (0.786 + 1.82i)59-s + (0.512 − 1.71i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $0.987 - 0.154i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (2095, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ 0.987 - 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9392916971\)
\(L(\frac12)\) \(\approx\) \(0.9392916971\)
\(L(1)\) 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 \( 1 + (0.597 + 0.802i)T \)
good5 \( 1 + (-0.973 - 0.230i)T^{2} \)
7 \( 1 + (-0.597 + 0.802i)T^{2} \)
11 \( 1 + (-0.0460 + 0.106i)T + (-0.686 - 0.727i)T^{2} \)
13 \( 1 + (-0.893 + 0.448i)T^{2} \)
17 \( 1 + (-0.337 - 1.91i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.137 - 0.780i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.597 - 0.802i)T^{2} \)
29 \( 1 + (0.835 - 0.549i)T^{2} \)
31 \( 1 + (-0.396 + 0.918i)T^{2} \)
37 \( 1 + (-0.766 - 0.642i)T^{2} \)
41 \( 1 + (0.819 + 0.868i)T + (-0.0581 + 0.998i)T^{2} \)
43 \( 1 + (-0.819 - 1.10i)T + (-0.286 + 0.957i)T^{2} \)
47 \( 1 + (-0.396 - 0.918i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.786 - 1.82i)T + (-0.686 + 0.727i)T^{2} \)
61 \( 1 + (0.993 + 0.116i)T^{2} \)
67 \( 1 + (-0.512 + 1.71i)T + (-0.835 - 0.549i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.439 + 0.368i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.0581 + 0.998i)T^{2} \)
83 \( 1 + (1.28 - 1.36i)T + (-0.0581 - 0.998i)T^{2} \)
89 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-1.65 + 0.193i)T + (0.973 - 0.230i)T^{2} \)
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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

−8.821442336628995496926311984877, −8.271541182096357355906110340082, −7.53975254509589453198330511603, −6.72679616824778150224026419878, −6.01040722231592173065132071140, −5.44864440471310552165201677536, −4.37203198736942052019872814252, −3.40577786604241121875973069584, −2.14517892764146029379054609701, −1.22130705117337089382326948043, 0.77306735872262284253327228411, 2.57032314912992248769615449111, 3.39453598045781682522020177914, 4.52175214043634962110333740461, 5.01093234909434954846078771159, 5.80838635900795910581256318586, 6.80586837437457416886210223439, 7.29708010190163987212544132779, 8.532234407179332420596726056773, 9.145803596643209696807821229138

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