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Properties

Label 2-864-72.11-c1-0-5
Degree $2$
Conductor $864$
Sign $0.700 + 0.713i$
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  + (−0.895 − 1.55i)5-s + (2.08 + 1.20i)7-s + (−1.36 − 0.790i)11-s + (5.35 − 3.09i)13-s + 3.69i·17-s − 3.12·19-s + (−1.36 − 2.35i)23-s + (0.896 − 1.55i)25-s + (2.55 − 4.42i)29-s + (5.95 − 3.43i)31-s − 4.31i·35-s + 5.24i·37-s + (5.32 − 3.07i)41-s + (0.452 − 0.783i)43-s + (4.88 − 8.46i)47-s + ⋯
L(s)  = 1  + (−0.400 − 0.693i)5-s + (0.789 + 0.455i)7-s + (−0.412 − 0.238i)11-s + (1.48 − 0.857i)13-s + 0.897i·17-s − 0.717·19-s + (−0.283 − 0.491i)23-s + (0.179 − 0.310i)25-s + (0.474 − 0.821i)29-s + (1.06 − 0.617i)31-s − 0.729i·35-s + 0.861i·37-s + (0.831 − 0.479i)41-s + (0.0689 − 0.119i)43-s + (0.713 − 1.23i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.700 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46199 - 0.613590i\)
\(L(\frac12)\) \(\approx\) \(1.46199 - 0.613590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.895 + 1.55i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.08 - 1.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.36 + 0.790i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.35 + 3.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.69iT - 17T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
23 \( 1 + (1.36 + 2.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.55 + 4.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.95 + 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.24iT - 37T^{2} \)
41 \( 1 + (-5.32 + 3.07i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.452 + 0.783i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.88 + 8.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.05T + 53T^{2} \)
59 \( 1 + (6.10 - 3.52i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.05 + 1.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.03 + 1.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.31T + 71T^{2} \)
73 \( 1 - 0.631T + 73T^{2} \)
79 \( 1 + (7.82 + 4.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.5 - 7.82i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.16iT - 89T^{2} \)
97 \( 1 + (6.72 - 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.28939420724477650034140917365, −8.877747965419978682663210685971, −8.269155543679860225835650720053, −8.001139799866527783749097618482, −6.38950762331984734963074891103, −5.70256764683152006886588905435, −4.65149750603497527187710452812, −3.80698010878224349803744831040, −2.36808387569057283902430706220, −0.903400282097783585186406573534, 1.36804290151906425312233410114, 2.82536900019274867268142216316, 3.97199800654426282021842898348, 4.77791079368304432516264770692, 6.04020177694860182892032795946, 6.94501611589746615974609378192, 7.65609093104308507773358759466, 8.532816602502810369517542767334, 9.369785230842706522208800682747, 10.59468472415652896916303873882

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