Properties

Label 2-864-36.7-c0-0-0
Degree $2$
Conductor $864$
Sign $0.906 + 0.422i$
Analytic cond. $0.431192$
Root an. cond. $0.656652$
Motivic weight $0$
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 + (0.866 + 0.5i)7-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s − 0.999i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s − 0.999i·55-s + (−0.866 + 0.5i)59-s + (−0.5 + 0.866i)61-s + (−0.499 + 0.866i)65-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s − 0.999i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s − 0.999i·55-s + (−0.866 + 0.5i)59-s + (−0.5 + 0.866i)61-s + (−0.499 + 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :0),\ 0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021080099\)
\(L(\frac12)\) \(\approx\) \(1.021080099\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.29187722659762269478218337853, −9.306824498823252272822129443780, −8.592635933702081706605653054265, −7.930696489816005994717155127207, −7.01676895785692401398037104609, −5.79422202109496432310981615385, −4.85322375233883259677854437046, −4.25470456806954043982600474452, −2.75698986940459864521023111760, −1.30070514683736634751888309100, 1.60253926696782985695749740993, 3.12158720315593732906587942993, 4.05484096150598663645884956589, 4.99342168953556612427332942731, 6.27570802743728072246292887696, 7.15978133256761823722844758022, 7.64406817390733957818753947442, 8.821766883617538511277111455022, 9.487845137820083376106443793033, 10.81126899436467808296038556099

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