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Properties

Label 2-864-36.11-c1-0-1
Degree $2$
Conductor $864$
Sign $-0.626 - 0.779i$
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  + (−1.81 + 1.04i)5-s + (0.143 + 0.0829i)7-s + (0.784 − 1.35i)11-s + (−1.93 − 3.34i)13-s + 5.27i·17-s + 8.05i·19-s + (2.67 + 4.63i)23-s + (−0.298 + 0.516i)25-s + (−6.75 − 3.89i)29-s + (−2.10 + 1.21i)31-s − 0.348·35-s − 8.53·37-s + (−2.47 + 1.43i)41-s + (−3.42 − 1.97i)43-s + (−3.68 + 6.38i)47-s + ⋯
L(s)  = 1  + (−0.812 + 0.469i)5-s + (0.0543 + 0.0313i)7-s + (0.236 − 0.409i)11-s + (−0.535 − 0.928i)13-s + 1.27i·17-s + 1.84i·19-s + (0.557 + 0.966i)23-s + (−0.0596 + 0.103i)25-s + (−1.25 − 0.724i)29-s + (−0.378 + 0.218i)31-s − 0.0588·35-s − 1.40·37-s + (−0.387 + 0.223i)41-s + (−0.521 − 0.301i)43-s + (−0.537 + 0.931i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.626 - 0.779i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305725 + 0.637787i\)
\(L(\frac12)\) \(\approx\) \(0.305725 + 0.637787i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.81 - 1.04i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.143 - 0.0829i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.784 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.93 + 3.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.27iT - 17T^{2} \)
19 \( 1 - 8.05iT - 19T^{2} \)
23 \( 1 + (-2.67 - 4.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.75 + 3.89i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.10 - 1.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.53T + 37T^{2} \)
41 \( 1 + (2.47 - 1.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.42 + 1.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.68 - 6.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.40iT - 53T^{2} \)
59 \( 1 + (-5.49 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.11 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.45 + 0.841i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (6.31 + 3.64i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.35 + 2.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.40iT - 89T^{2} \)
97 \( 1 + (-0.903 + 1.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.47509087009774374278153840110, −9.791668936065031369641329378648, −8.584514576351717964006233774469, −7.88649552924365948900770828750, −7.25119768481346904623849459374, −6.05221757202415264345471683709, −5.31658488315028031537451213497, −3.80230063815657345834230201258, −3.39229887158456488939316601501, −1.69991860332330337703728626190, 0.33599748524413812670043274571, 2.13684585800862854127235328180, 3.45795223066490395799017081288, 4.68853425113025401774430832241, 5.02445635975803717269161046150, 6.81261407858670863181508407757, 7.08891855781848595954420779032, 8.225481384964708098073124639115, 9.127568815532556942568703978733, 9.579652401827903697580959025483

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