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Properties

Label 2-864-216.133-c1-0-32
Degree $2$
Conductor $864$
Sign $-0.958 - 0.286i$
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  + (−0.411 − 1.68i)3-s + (−2.15 − 2.57i)5-s + (1.78 − 0.649i)7-s + (−2.66 + 1.38i)9-s + (3.21 − 3.83i)11-s + (−3.61 + 0.637i)13-s + (−3.44 + 4.69i)15-s + (−1.74 − 3.02i)17-s + (−2.73 − 1.57i)19-s + (−1.82 − 2.73i)21-s + (2.93 + 1.06i)23-s + (−1.09 + 6.19i)25-s + (3.42 + 3.90i)27-s + (6.61 + 1.16i)29-s + (−0.842 − 0.306i)31-s + ⋯
L(s)  = 1  + (−0.237 − 0.971i)3-s + (−0.965 − 1.15i)5-s + (0.674 − 0.245i)7-s + (−0.887 + 0.461i)9-s + (0.970 − 1.15i)11-s + (−1.00 + 0.176i)13-s + (−0.888 + 1.21i)15-s + (−0.422 − 0.732i)17-s + (−0.626 − 0.361i)19-s + (−0.398 − 0.596i)21-s + (0.611 + 0.222i)23-s + (−0.218 + 1.23i)25-s + (0.658 + 0.752i)27-s + (1.22 + 0.216i)29-s + (−0.151 − 0.0550i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.958 - 0.286i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116370 + 0.795321i\)
\(L(\frac12)\) \(\approx\) \(0.116370 + 0.795321i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.411 + 1.68i)T \)
good5 \( 1 + (2.15 + 2.57i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.78 + 0.649i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-3.21 + 3.83i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (3.61 - 0.637i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.74 + 3.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.73 + 1.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.93 - 1.06i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-6.61 - 1.16i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.842 + 0.306i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (6.21 - 3.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.341 + 1.93i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.36 - 6.39i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.63 - 0.959i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 9.52iT - 53T^{2} \)
59 \( 1 + (6.13 + 7.31i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.92 + 8.03i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-13.7 + 2.42i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.50 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.472 + 0.818i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.803 + 4.55i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.12 - 0.375i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.83 - 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.68 - 1.41i)T + (16.8 + 95.5i)T^{2} \)
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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

−9.411375790227035088246600361924, −8.562628986641509978033535845006, −8.145672126739108166898087427097, −7.17835025624871838274357745296, −6.40484102666425839691324343538, −5.06012653651686125038054113124, −4.55689268397352604602298010838, −3.11833509106243189277144143826, −1.49662791682779054318400894086, −0.41395519577965874372447273527, 2.23813087120140180335033408290, 3.50682840818150018122910388006, 4.31401460392263148200752181096, 5.07515807561606888858855602696, 6.48338541596036689644378299947, 7.09056381090829159194752380559, 8.163830517368395223374458344693, 8.949015904105462421775034356788, 10.11549396001975003108488379829, 10.46700724041361185877195866675

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