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Properties

Label 2-864-12.11-c1-0-2
Degree $2$
Conductor $864$
Sign $-0.707 - 0.707i$
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  + 3.14i·5-s + 3.44i·7-s + 4.56·11-s − 6.89·13-s − 3.46i·17-s + 4.89i·19-s − 2.82·23-s − 4.89·25-s − 2.19i·29-s − 2.55i·31-s − 10.8·35-s − 4.89·37-s − 4.09i·41-s + 2.89i·43-s + 2.19·47-s + ⋯
L(s)  = 1  + 1.40i·5-s + 1.30i·7-s + 1.37·11-s − 1.91·13-s − 0.840i·17-s + 1.12i·19-s − 0.589·23-s − 0.979·25-s − 0.407i·29-s − 0.458i·31-s − 1.83·35-s − 0.805·37-s − 0.640i·41-s + 0.442i·43-s + 0.319·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467821 + 1.12941i\)
\(L(\frac12)\) \(\approx\) \(0.467821 + 1.12941i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.14iT - 5T^{2} \)
7 \( 1 - 3.44iT - 7T^{2} \)
11 \( 1 - 4.56T + 11T^{2} \)
13 \( 1 + 6.89T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 2.19iT - 29T^{2} \)
31 \( 1 + 2.55iT - 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 + 4.09iT - 41T^{2} \)
43 \( 1 - 2.89iT - 43T^{2} \)
47 \( 1 - 2.19T + 47T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 + 2.19T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 7.89T + 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 5.02iT - 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−10.32858205629930773466001823026, −9.676670755205154428824150348031, −8.991975797528930515684431799743, −7.76352523215060245891767810768, −7.02983542691537277647982988671, −6.22224854718811986050652399767, −5.36814115552168201857856597648, −4.05912873788872775961289346324, −2.86151156763572732529003026655, −2.13250426643098479650449862743, 0.58130668972648812874663419836, 1.81558208441337777273868333513, 3.64988230787385998315288219316, 4.54281437841194042920993189821, 5.10602363516877063251773068688, 6.56111642941403320173182703872, 7.25641215247508720793210791142, 8.211814493268430091102642039164, 9.102446272260269141352831739068, 9.722687747477033497504213161308

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