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Properties

Label 2-864-288.131-c1-0-0
Degree $2$
Conductor $864$
Sign $-0.567 - 0.823i$
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  + (0.0101 − 1.41i)2-s + (−1.99 − 0.0287i)4-s + (−0.203 + 1.54i)5-s + (0.0488 − 0.182i)7-s + (−0.0609 + 2.82i)8-s + (2.18 + 0.303i)10-s + (1.65 + 1.27i)11-s + (−3.11 − 4.06i)13-s + (−0.257 − 0.0708i)14-s + (3.99 + 0.114i)16-s − 6.30·17-s + (−4.50 − 1.86i)19-s + (0.451 − 3.08i)20-s + (1.81 − 2.32i)22-s + (−4.54 + 1.21i)23-s + ⋯
L(s)  = 1  + (0.00718 − 0.999i)2-s + (−0.999 − 0.0143i)4-s + (−0.0910 + 0.691i)5-s + (0.0184 − 0.0688i)7-s + (−0.0215 + 0.999i)8-s + (0.691 + 0.0960i)10-s + (0.499 + 0.382i)11-s + (−0.864 − 1.12i)13-s + (−0.0687 − 0.0189i)14-s + (0.999 + 0.0287i)16-s − 1.52·17-s + (−1.03 − 0.428i)19-s + (0.101 − 0.690i)20-s + (0.386 − 0.496i)22-s + (−0.947 + 0.253i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00630571 + 0.0120090i\)
\(L(\frac12)\) \(\approx\) \(0.00630571 + 0.0120090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0101 + 1.41i)T \)
3 \( 1 \)
good5 \( 1 + (0.203 - 1.54i)T + (-4.82 - 1.29i)T^{2} \)
7 \( 1 + (-0.0488 + 0.182i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.65 - 1.27i)T + (2.84 + 10.6i)T^{2} \)
13 \( 1 + (3.11 + 4.06i)T + (-3.36 + 12.5i)T^{2} \)
17 \( 1 + 6.30T + 17T^{2} \)
19 \( 1 + (4.50 + 1.86i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.54 - 1.21i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.26 - 0.561i)T + (28.0 - 7.50i)T^{2} \)
31 \( 1 + (5.30 + 3.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.61 - 3.90i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.831 + 3.10i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.52 - 5.90i)T + (-11.1 - 41.5i)T^{2} \)
47 \( 1 + (1.00 - 0.582i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.04 - 7.35i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (9.10 + 1.19i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (0.850 + 6.46i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (1.34 + 1.74i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (-5.65 + 5.65i)T - 71iT^{2} \)
73 \( 1 + (-6.45 - 6.45i)T + 73iT^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.6 - 1.39i)T + (80.1 - 21.4i)T^{2} \)
89 \( 1 + (-1.53 - 1.53i)T + 89iT^{2} \)
97 \( 1 + (-4.59 - 7.96i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.69547721727273295469733770049, −9.734383566149021038199803300125, −9.039555390156600310974906797177, −8.035776832589843239900010600656, −7.11818927231726475037842992244, −6.05493297841015198239245052118, −4.85082818965368168143965351388, −4.01590912343286313377743848859, −2.87915269629640554519845475349, −1.97113260227995027470483631016, 0.00620386875612544634958835933, 1.97446373759546815288654316457, 3.95355429822534920576264463831, 4.50719059150348146963211293958, 5.53058773315874105827052129550, 6.54587516349740531807430048172, 7.12472038702990183603782531574, 8.343394028382495372964092149183, 8.853435692210577970663224651595, 9.484507512623938094555098022587

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