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Properties

Label 2-864-36.11-c1-0-5
Degree $2$
Conductor $864$
Sign $0.951 + 0.308i$
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  + (0.135 − 0.0779i)5-s + (0.349 + 0.201i)7-s + (2.28 − 3.95i)11-s + (2.14 + 3.71i)13-s − 2.84i·17-s + 0.958i·19-s + (−2.71 − 4.69i)23-s + (−2.48 + 4.30i)25-s + (4.88 + 2.82i)29-s + (8.53 − 4.92i)31-s + 0.0628·35-s + 9.89·37-s + (7.79 − 4.49i)41-s + (−2.59 − 1.49i)43-s + (−1.22 + 2.12i)47-s + ⋯
L(s)  = 1  + (0.0603 − 0.0348i)5-s + (0.132 + 0.0762i)7-s + (0.689 − 1.19i)11-s + (0.594 + 1.02i)13-s − 0.688i·17-s + 0.219i·19-s + (−0.565 − 0.979i)23-s + (−0.497 + 0.861i)25-s + (0.907 + 0.524i)29-s + (1.53 − 0.885i)31-s + 0.0106·35-s + 1.62·37-s + (1.21 − 0.702i)41-s + (−0.395 − 0.228i)43-s + (−0.178 + 0.309i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67917 - 0.265800i\)
\(L(\frac12)\) \(\approx\) \(1.67917 - 0.265800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.135 + 0.0779i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.349 - 0.201i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 3.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.14 - 3.71i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.84iT - 17T^{2} \)
19 \( 1 - 0.958iT - 19T^{2} \)
23 \( 1 + (2.71 + 4.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.88 - 2.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.53 + 4.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 + (-7.79 + 4.49i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.59 + 1.49i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.22 - 2.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.45iT - 53T^{2} \)
59 \( 1 + (1.18 + 2.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.10 - 7.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 + 6.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 4.53T + 73T^{2} \)
79 \( 1 + (2.84 + 1.64i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.812 + 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.45iT - 89T^{2} \)
97 \( 1 + (0.162 - 0.281i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.05812466624326176078633592363, −9.179544212361459561578801050474, −8.558724711820859621508388777055, −7.65103812726655742142979568812, −6.44134620593099076668227830315, −6.00323937769812551252878544173, −4.65844568109921102954653542256, −3.79149768111182210498469568817, −2.55656151419820258680036255568, −1.04292551517125190473493376155, 1.26047146375726138356560208273, 2.64123617399922298023269284783, 3.93886188835718395694109307597, 4.75405242184313181591887701222, 5.99748966094556249183225452361, 6.63764967642207948128862164810, 7.86614285588798391048353875391, 8.303227278966474842285045442088, 9.639037967278366802742952481381, 10.00821238521100294358681352825

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