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Properties

Label 2-864-9.7-c1-0-9
Degree $2$
Conductor $864$
Sign $-0.569 + 0.821i$
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  + (−0.5 − 0.866i)5-s + (−1.72 + 2.98i)7-s + (0.724 − 1.25i)11-s + (−2.94 − 5.10i)13-s − 4.89·17-s + 4·19-s + (−2.72 − 4.71i)23-s + (2 − 3.46i)25-s + (0.0505 − 0.0874i)29-s + (−1.27 − 2.20i)31-s + 3.44·35-s − 0.898·37-s + (−5.94 − 10.3i)41-s + (1.17 − 2.03i)43-s + (−3.17 + 5.49i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.651 + 1.12i)7-s + (0.218 − 0.378i)11-s + (−0.818 − 1.41i)13-s − 1.18·17-s + 0.917·19-s + (−0.568 − 0.984i)23-s + (0.400 − 0.692i)25-s + (0.00937 − 0.0162i)29-s + (−0.229 − 0.396i)31-s + 0.583·35-s − 0.147·37-s + (−0.929 − 1.60i)41-s + (0.179 − 0.310i)43-s + (−0.463 + 0.801i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.569 + 0.821i$
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: \((2,\ 864,\ (\ :1/2),\ -0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304823 - 0.582277i\)
\(L(\frac12)\) \(\approx\) \(0.304823 - 0.582277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.72 - 2.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 + 5.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (2.72 + 4.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0505 + 0.0874i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.27 + 2.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.898T + 37T^{2} \)
41 \( 1 + (5.94 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.17 + 2.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.17 - 5.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (-7.17 - 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.94 + 6.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.79T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 + (6.72 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.275 - 0.476i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-1.94 + 3.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.831044195531228039085058486888, −8.922127558919943748868952935155, −8.386050332662795934259830668503, −7.34753050017062907001220216539, −6.26910103828568059464996969194, −5.51778099768145108919778639558, −4.58262628462686661984446099278, −3.22597956849282610321532193354, −2.34428219328039793434820109032, −0.30489569258622436693407985971, 1.69127943503867575189940154300, 3.18310854133402948892821618599, 4.10702458622560280148647835653, 4.97194291425726780116558570849, 6.48682576550809438461475936852, 7.00613372952920521259098575205, 7.62400378131771714067984262620, 8.975517433074412150947352715228, 9.691922317965669556174486157419, 10.30265823398956368450282578035

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