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Properties

Label 2-864-9.7-c1-0-2
Degree $2$
Conductor $864$
Sign $0.747 - 0.664i$
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  + (−1.68 − 2.92i)5-s + (−2.35 + 4.07i)7-s + (0.437 − 0.758i)11-s + (0.686 + 1.18i)13-s + 2.37·17-s + 5.57·19-s + (2.35 + 4.07i)23-s + (−3.18 + 5.51i)25-s + (−2.68 + 4.65i)29-s + (3.22 + 5.58i)31-s + 15.8·35-s + 4·37-s + (0.5 + 0.866i)41-s + (0.437 − 0.758i)43-s + (2.35 − 4.07i)47-s + ⋯
L(s)  = 1  + (−0.754 − 1.30i)5-s + (−0.888 + 1.53i)7-s + (0.131 − 0.228i)11-s + (0.190 + 0.329i)13-s + 0.575·17-s + 1.27·19-s + (0.490 + 0.849i)23-s + (−0.637 + 1.10i)25-s + (−0.498 + 0.863i)29-s + (0.579 + 1.00i)31-s + 2.68·35-s + 0.657·37-s + (0.0780 + 0.135i)41-s + (0.0667 − 0.115i)43-s + (0.342 − 0.594i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.747 - 0.664i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06041 + 0.402927i\)
\(L(\frac12)\) \(\approx\) \(1.06041 + 0.402927i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 2.92i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.35 - 4.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.437 + 0.758i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.686 - 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 + (-2.35 - 4.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.68 - 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.437 + 0.758i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.35 + 4.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (-4.26 - 7.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.26 - 7.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.47 - 2.55i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
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

−9.996407738961643214854312381322, −9.132872853123878948523680776720, −8.824195981733304479886052323569, −7.898880118876996208736354635062, −6.83876177604981203589152350187, −5.57853455435230046560953836412, −5.21305315494272559200299891305, −3.83715546208704768284211645110, −2.89210789538956802342472331822, −1.19055159438007230825330408183, 0.66054455953531869808678988780, 2.84095004152413105553291356245, 3.58099988778297747620797619182, 4.36097060753509450556304520062, 5.95085635332379235752888568292, 6.84644970176753768148412689047, 7.40039129303023568822250655889, 8.017611842983589648156630841099, 9.608568184666565964187952333479, 10.06700702705324039595180491937

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