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Properties

Label 2-864-27.25-c1-0-9
Degree $2$
Conductor $864$
Sign $-0.0825 - 0.996i$
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  + (−0.882 + 1.49i)3-s + (1.89 − 1.58i)5-s + (−2.25 + 0.819i)7-s + (−1.44 − 2.63i)9-s + (0.591 + 0.496i)11-s + (0.133 + 0.755i)13-s + (0.696 + 4.22i)15-s + (3.04 + 5.27i)17-s + (−2.64 + 4.58i)19-s + (0.766 − 4.07i)21-s + (0.689 + 0.251i)23-s + (0.193 − 1.09i)25-s + (5.19 + 0.175i)27-s + (−0.0924 + 0.524i)29-s + (8.58 + 3.12i)31-s + ⋯
L(s)  = 1  + (−0.509 + 0.860i)3-s + (0.847 − 0.710i)5-s + (−0.850 + 0.309i)7-s + (−0.480 − 0.877i)9-s + (0.178 + 0.149i)11-s + (0.0369 + 0.209i)13-s + (0.179 + 1.09i)15-s + (0.738 + 1.27i)17-s + (−0.607 + 1.05i)19-s + (0.167 − 0.889i)21-s + (0.143 + 0.0523i)23-s + (0.0387 − 0.219i)25-s + (0.999 + 0.0337i)27-s + (−0.0171 + 0.0974i)29-s + (1.54 + 0.560i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0825 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0825 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.0825 - 0.996i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.0825 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.806136 + 0.875674i\)
\(L(\frac12)\) \(\approx\) \(0.806136 + 0.875674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.882 - 1.49i)T \)
good5 \( 1 + (-1.89 + 1.58i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (2.25 - 0.819i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-0.591 - 0.496i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.133 - 0.755i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.64 - 4.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.689 - 0.251i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.0924 - 0.524i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-8.58 - 3.12i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-0.587 - 1.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.88 - 10.6i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.29 + 4.44i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-4.78 + 1.74i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 9.59T + 53T^{2} \)
59 \( 1 + (7.85 - 6.59i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-4.38 + 1.59i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.191 + 1.08i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-4.00 - 6.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.53 + 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.56 - 14.5i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.03 + 11.5i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (2.82 - 4.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.72 + 2.28i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.04996350156125500318581802542, −9.807649379134304602226091696171, −8.903792725296982430705786918008, −8.136385029441776764547898056439, −6.43828184942877739484794482059, −6.06437136108259332080090868972, −5.15759823791297114545018040460, −4.16274491298763017660841462634, −3.12202072929620204963155749304, −1.45797355004152994244199930045, 0.64167889816941100242691262375, 2.31878861131930252608025173736, 3.12270728059526456782089320146, 4.78561104017142785854548724919, 5.85467542358730419430345963180, 6.54219760175491619114897942531, 7.07874045834883505105267292487, 8.067635036891548785205730026580, 9.283492507682035505493601694845, 10.00155552808791262332898936070

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