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Properties

Label 2-864-72.29-c2-0-12
Degree $2$
Conductor $864$
Sign $0.958 + 0.285i$
Analytic cond. $23.5422$
Root an. cond. $4.85204$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.693 − 1.20i)5-s + (0.562 − 0.975i)7-s + (−1.11 + 1.92i)11-s + (14.4 − 8.34i)13-s + 20.2i·17-s + 21.0i·19-s + (−18.0 + 10.4i)23-s + (11.5 − 19.9i)25-s + (26.9 − 46.7i)29-s + (−9.19 − 15.9i)31-s − 1.56·35-s − 34.8i·37-s + (15.1 − 8.72i)41-s + (44.2 + 25.5i)43-s + (32.4 + 18.7i)47-s + ⋯
L(s)  = 1  + (−0.138 − 0.240i)5-s + (0.0804 − 0.139i)7-s + (−0.101 + 0.175i)11-s + (1.11 − 0.642i)13-s + 1.19i·17-s + 1.10i·19-s + (−0.784 + 0.452i)23-s + (0.461 − 0.799i)25-s + (0.929 − 1.61i)29-s + (−0.296 − 0.513i)31-s − 0.0445·35-s − 0.941i·37-s + (0.368 − 0.212i)41-s + (1.02 + 0.593i)43-s + (0.690 + 0.398i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ 0.958 + 0.285i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.894046249\)
\(L(\frac12)\) \(\approx\) \(1.894046249\)
\(L(2)\) 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 \( 1 + (0.693 + 1.20i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-0.562 + 0.975i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (1.11 - 1.92i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-14.4 + 8.34i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 20.2iT - 289T^{2} \)
19 \( 1 - 21.0iT - 361T^{2} \)
23 \( 1 + (18.0 - 10.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-26.9 + 46.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (9.19 + 15.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 34.8iT - 1.36e3T^{2} \)
41 \( 1 + (-15.1 + 8.72i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-44.2 - 25.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-32.4 - 18.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 52.6T + 2.80e3T^{2} \)
59 \( 1 + (27.1 + 46.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-76.5 - 44.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-66.0 + 38.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 69.4iT - 5.04e3T^{2} \)
73 \( 1 + 82.6T + 5.32e3T^{2} \)
79 \( 1 + (19.8 - 34.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-56.9 + 98.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 25.3iT - 7.92e3T^{2} \)
97 \( 1 + (-10.4 + 18.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.12440979374193284696797458607, −8.986693996301081414133303842524, −8.131131119824288675225635893935, −7.66167411153196021494767279250, −6.14940301536555899844761840071, −5.83197198566173460971495157524, −4.33149371209727594447070570899, −3.70190867364060698874717647500, −2.21959462042327369724119666147, −0.860550752494021529773981544729, 0.968153955748505716668575389588, 2.49875905720797729205305345920, 3.53494210935663968383301722289, 4.67200499657041223224404083529, 5.58281557080278438691371982404, 6.75437953816008604862299990008, 7.22291069319219069823515870764, 8.622765852585927118876483745711, 8.905859071668812141186366355341, 10.06143895773612329984510803501

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