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Properties

Label 2-864-8.5-c1-0-14
Degree $2$
Conductor $864$
Sign $-0.210 + 0.977i$
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  − 3.36i·5-s + 2.64·7-s − 2.16i·11-s − 4.64i·13-s − 4.55·17-s + 6.29i·19-s + 0.979·23-s − 6.29·25-s − 4.33i·29-s − 2·31-s − 8.89i·35-s + 1.35i·37-s + 11.0·41-s − 3.29i·43-s − 10.0·47-s + ⋯
L(s)  = 1  − 1.50i·5-s + 0.999·7-s − 0.654i·11-s − 1.28i·13-s − 1.10·17-s + 1.44i·19-s + 0.204·23-s − 1.25·25-s − 0.805i·29-s − 0.359·31-s − 1.50i·35-s + 0.222i·37-s + 1.72·41-s − 0.501i·43-s − 1.47·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.210 + 0.977i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.210 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.958673 - 1.18698i\)
\(L(\frac12)\) \(\approx\) \(0.958673 - 1.18698i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.36iT - 5T^{2} \)
7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 + 4.55T + 17T^{2} \)
19 \( 1 - 6.29iT - 19T^{2} \)
23 \( 1 - 0.979T + 23T^{2} \)
29 \( 1 + 4.33iT - 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 1.35iT - 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 3.29iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 4.33iT - 53T^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 - 1.93iT - 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 8.64T + 79T^{2} \)
83 \( 1 - 2.38iT - 83T^{2} \)
89 \( 1 - 4.55T + 89T^{2} \)
97 \( 1 - 2.29T + 97T^{2} \)
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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

−9.833360624985843841490545468932, −8.928830512420312949340755958927, −8.128769160090417322015417261842, −7.87992091462239408625037780104, −6.22537588727300168952546345859, −5.36015144835645649965817428667, −4.71150293664521558632924089633, −3.64704220285438387175083654345, −1.98127223175644880455690087860, −0.75138378105902760509960331117, 1.89886687837026798273109343862, 2.77589944154465405388693319393, 4.18120994562881468883125064487, 4.92415084583555177356615801286, 6.38301668007006697708279620471, 6.96727921750992552023637515293, 7.60823363684724201043452331077, 8.845671015186013296690506249834, 9.512372836056905801213304818683, 10.68957285718880943114686859516

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