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Properties

Label 2-864-108.11-c1-0-20
Degree $2$
Conductor $864$
Sign $-0.558 + 0.829i$
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  + (−0.903 − 1.47i)3-s + (−1.26 + 3.46i)5-s + (−1.82 − 0.321i)7-s + (−1.36 + 2.67i)9-s + (2.49 − 0.907i)11-s + (4.15 − 3.48i)13-s + (6.25 − 1.26i)15-s + (−5.06 + 2.92i)17-s + (−6.05 − 3.49i)19-s + (1.17 + 2.98i)21-s + (−0.349 − 1.98i)23-s + (−6.56 − 5.51i)25-s + (5.18 − 0.397i)27-s + (4.64 − 5.53i)29-s + (5.28 − 0.932i)31-s + ⋯
L(s)  = 1  + (−0.521 − 0.852i)3-s + (−0.563 + 1.54i)5-s + (−0.689 − 0.121i)7-s + (−0.455 + 0.890i)9-s + (0.751 − 0.273i)11-s + (1.15 − 0.966i)13-s + (1.61 − 0.327i)15-s + (−1.22 + 0.709i)17-s + (−1.38 − 0.802i)19-s + (0.256 + 0.651i)21-s + (−0.0728 − 0.413i)23-s + (−1.31 − 1.10i)25-s + (0.997 − 0.0764i)27-s + (0.861 − 1.02i)29-s + (0.950 − 0.167i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.558 + 0.829i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.558 + 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245840 - 0.461881i\)
\(L(\frac12)\) \(\approx\) \(0.245840 - 0.461881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.903 + 1.47i)T \)
good5 \( 1 + (1.26 - 3.46i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.82 + 0.321i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.49 + 0.907i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-4.15 + 3.48i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (5.06 - 2.92i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.05 + 3.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.349 + 1.98i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-4.64 + 5.53i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-5.28 + 0.932i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.56 + 2.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.45 + 6.49i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.31 + 9.11i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.04 - 5.94i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 2.61iT - 53T^{2} \)
59 \( 1 + (-1.31 - 0.478i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.08 - 6.15i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.80 + 8.10i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (3.21 + 5.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.960 - 1.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.70 - 5.61i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.31 + 1.10i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-8.51 - 4.91i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.07 + 1.84i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.44599857696345591228215432213, −8.748160613012153307401605322786, −8.132914091745226648555908293976, −6.96545343789608796915293030890, −6.46799175153895341062027952457, −6.06485872871871578668289311095, −4.28819530177831772964776729541, −3.28133936514324243950019914465, −2.24384450464058189214626298359, −0.28464677093538077527644193519, 1.37372418938412982955869728242, 3.46332777599866513539792279998, 4.41192545243271347160654605608, 4.77223824018662781554680242786, 6.19478364576523270010948412301, 6.65903337865110110114786307631, 8.424670767140249513132653227648, 8.798414409003468127397292085561, 9.468007112011887778595205796817, 10.36577260151537111489915035445

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