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Properties

Label 2-72e2-24.11-c1-0-76
Degree $2$
Conductor $5184$
Sign $0.965 + 0.258i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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  + 4.04·5-s + 3.89i·7-s − 0.468i·11-s − 4.89i·13-s − 3.57i·17-s − 1.89·19-s + 8.56·23-s + 11.3·25-s − 2.36·29-s − 7.18i·31-s + 15.7i·35-s − 6.16i·37-s − 5.13i·41-s − 10.5·43-s + 9.37·47-s + ⋯
L(s)  = 1  + 1.80·5-s + 1.47i·7-s − 0.141i·11-s − 1.35i·13-s − 0.867i·17-s − 0.435·19-s + 1.78·23-s + 2.27·25-s − 0.439·29-s − 1.28i·31-s + 2.66i·35-s − 1.01i·37-s − 0.801i·41-s − 1.60·43-s + 1.36·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.965 + 0.258i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.970040693\)
\(L(\frac12)\) \(\approx\) \(2.970040693\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.04T + 5T^{2} \)
7 \( 1 - 3.89iT - 7T^{2} \)
11 \( 1 + 0.468iT - 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 + 3.57iT - 17T^{2} \)
19 \( 1 + 1.89T + 19T^{2} \)
23 \( 1 - 8.56T + 23T^{2} \)
29 \( 1 + 2.36T + 29T^{2} \)
31 \( 1 + 7.18iT - 31T^{2} \)
37 \( 1 + 6.16iT - 37T^{2} \)
41 \( 1 + 5.13iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 9.37T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 + 6.26iT - 59T^{2} \)
61 \( 1 + 0.983iT - 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 3.02T + 71T^{2} \)
73 \( 1 - 4.58T + 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + 5.66iT - 83T^{2} \)
89 \( 1 - 3.17iT - 89T^{2} \)
97 \( 1 - 0.611T + 97T^{2} \)
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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

−8.476712901151108289424468190631, −7.37163130584375319950900301314, −6.58350726759647394407749058359, −5.75996032202090777014551354588, −5.46617078027230086147212725697, −4.94348296122514984160251444141, −3.38631198176706170706312390711, −2.49247258030340431590569405106, −2.19257093701751810019228929889, −0.834227260030314766597297739188, 1.20404778432676945936821468410, 1.70281048922341112274259226157, 2.78092386312246623024769015200, 3.80440763380192704787253577336, 4.66749620149250115171710431897, 5.26026974390277613571658878256, 6.29790003961313329102275199847, 6.76740040731968435117562318982, 7.18958278918857126729056752292, 8.420794940147371442242443904125

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