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Properties

Label 2-864-24.11-c1-0-13
Degree $2$
Conductor $864$
Sign $-0.769 + 0.639i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  − 1.59·5-s − 1.43i·7-s − 4.93i·11-s + 5.37i·13-s − 1.32i·17-s − 0.267·19-s − 5.94·23-s − 2.46·25-s − 8.70·29-s − 7.86i·31-s + 2.29i·35-s − 2.49i·37-s − 9.87i·41-s − 2·43-s − 9.12·47-s + ⋯
L(s)  = 1  − 0.712·5-s − 0.544i·7-s − 1.48i·11-s + 1.48i·13-s − 0.320i·17-s − 0.0614·19-s − 1.23·23-s − 0.492·25-s − 1.61·29-s − 1.41i·31-s + 0.387i·35-s − 0.409i·37-s − 1.54i·41-s − 0.304·43-s − 1.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.769 + 0.639i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.769 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202371 - 0.560153i\)
\(L(\frac12)\) \(\approx\) \(0.202371 - 0.560153i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.59T + 5T^{2} \)
7 \( 1 + 1.43iT - 7T^{2} \)
11 \( 1 + 4.93iT - 11T^{2} \)
13 \( 1 - 5.37iT - 13T^{2} \)
17 \( 1 + 1.32iT - 17T^{2} \)
19 \( 1 + 0.267T + 19T^{2} \)
23 \( 1 + 5.94T + 23T^{2} \)
29 \( 1 + 8.70T + 29T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + 2.49iT - 37T^{2} \)
41 \( 1 + 9.87iT - 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 9.12T + 47T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 + 4.93iT - 59T^{2} \)
61 \( 1 + 5.37iT - 61T^{2} \)
67 \( 1 + 1.19T + 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 7.92T + 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 + 7.23iT - 83T^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 - 9.39T + 97T^{2} \)
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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

−9.757242823077528035267023076897, −8.985903064747596258140515378027, −8.078853529956338989535015533997, −7.37790517907161817523048908940, −6.39372438271824781656960502072, −5.49760344186242005584745772683, −4.06016231566620089119447718855, −3.71372189108332701931025500994, −2.05728516653361004795692642416, −0.27663923914717988206676399577, 1.82151354683557743803334794129, 3.13358903098856928363560712587, 4.19349433497971501081409897833, 5.18668790086308589754858997350, 6.10071180583936959584774903531, 7.30326912961332435348231863161, 7.88152937523880048679875834445, 8.691937011392697034144964499900, 9.836675023813781316037457702029, 10.30512856326332654618661919926

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