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Properties

Label 2-864-27.22-c1-0-21
Degree $2$
Conductor $864$
Sign $0.816 - 0.577i$
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  + (0.418 + 1.68i)3-s + (3.54 − 1.28i)5-s + (−0.131 − 0.745i)7-s + (−2.64 + 1.40i)9-s + (5.73 + 2.08i)11-s + (−3.77 − 3.16i)13-s + (3.65 + 5.41i)15-s + (−0.681 + 1.17i)17-s + (2.71 + 4.69i)19-s + (1.19 − 0.533i)21-s + (0.406 − 2.30i)23-s + (7.06 − 5.93i)25-s + (−3.47 − 3.86i)27-s + (4.13 − 3.47i)29-s + (−0.295 + 1.67i)31-s + ⋯
L(s)  = 1  + (0.241 + 0.970i)3-s + (1.58 − 0.576i)5-s + (−0.0497 − 0.281i)7-s + (−0.883 + 0.469i)9-s + (1.72 + 0.629i)11-s + (−1.04 − 0.878i)13-s + (0.942 + 1.39i)15-s + (−0.165 + 0.286i)17-s + (0.622 + 1.07i)19-s + (0.261 − 0.116i)21-s + (0.0847 − 0.480i)23-s + (1.41 − 1.18i)25-s + (−0.668 − 0.743i)27-s + (0.767 − 0.644i)29-s + (−0.0530 + 0.301i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16880 + 0.690040i\)
\(L(\frac12)\) \(\approx\) \(2.16880 + 0.690040i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.418 - 1.68i)T \)
good5 \( 1 + (-3.54 + 1.28i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.131 + 0.745i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-5.73 - 2.08i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.77 + 3.16i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.681 - 1.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 - 4.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.406 + 2.30i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-4.13 + 3.47i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.295 - 1.67i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.11 + 1.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.22 + 1.02i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.88 + 2.14i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.93 - 10.9i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 7.06T + 53T^{2} \)
59 \( 1 + (11.6 - 4.24i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.963 - 5.46i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.0435 + 0.0365i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.01 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.29 - 7.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.50 - 2.94i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.43 - 4.55i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (4.12 + 7.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.65 - 0.964i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.997438344242141058134446602024, −9.552710359934240309629860587844, −8.934571220363648163436421795669, −7.88338222614273797519408127013, −6.56895650105983965117244191566, −5.75839365846208416377419481999, −4.90465822064544387561938003156, −4.04911904145124443158765347171, −2.71536171588015458615720114329, −1.47494043357045700686786663329, 1.35865237541174417693361727767, 2.31907974553476572053452800311, 3.26197987323986316296938946740, 4.97361085987709644224092512739, 6.02035055909103826661333514958, 6.70738653538906045151847338987, 7.10894997189294792613822995163, 8.608622996827019934984402441699, 9.346098353013577835556903965533, 9.691851507958696531483598584341

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