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Properties

Label 2-864-32.13-c1-0-24
Degree $2$
Conductor $864$
Sign $0.794 + 0.607i$
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  + (0.239 − 1.39i)2-s + (−1.88 − 0.666i)4-s + (−0.369 + 0.153i)5-s + (−1.24 + 1.24i)7-s + (−1.37 + 2.46i)8-s + (0.125 + 0.551i)10-s + (0.490 + 1.18i)11-s + (4.60 + 1.90i)13-s + (1.43 + 2.02i)14-s + (3.11 + 2.51i)16-s − 4.73i·17-s + (2.25 + 0.932i)19-s + (0.798 − 0.0423i)20-s + (1.76 − 0.400i)22-s + (4.41 + 4.41i)23-s + ⋯
L(s)  = 1  + (0.169 − 0.985i)2-s + (−0.942 − 0.333i)4-s + (−0.165 + 0.0684i)5-s + (−0.469 + 0.469i)7-s + (−0.487 + 0.872i)8-s + (0.0395 + 0.174i)10-s + (0.147 + 0.357i)11-s + (1.27 + 0.528i)13-s + (0.383 + 0.541i)14-s + (0.777 + 0.628i)16-s − 1.14i·17-s + (0.516 + 0.214i)19-s + (0.178 − 0.00947i)20-s + (0.376 − 0.0854i)22-s + (0.919 + 0.919i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37189 - 0.464535i\)
\(L(\frac12)\) \(\approx\) \(1.37189 - 0.464535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.239 + 1.39i)T \)
3 \( 1 \)
good5 \( 1 + (0.369 - 0.153i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.24 - 1.24i)T - 7iT^{2} \)
11 \( 1 + (-0.490 - 1.18i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-4.60 - 1.90i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.73iT - 17T^{2} \)
19 \( 1 + (-2.25 - 0.932i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \)
29 \( 1 + (-3.72 + 8.99i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 5.82T + 31T^{2} \)
37 \( 1 + (1.91 - 0.792i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.14 - 3.14i)T + 41iT^{2} \)
43 \( 1 + (-1.99 - 4.80i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.99iT - 47T^{2} \)
53 \( 1 + (0.855 + 2.06i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.65 + 0.683i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.408 + 0.987i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (4.01 - 9.70i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.17 - 2.17i)T - 71iT^{2} \)
73 \( 1 + (-7.91 - 7.91i)T + 73iT^{2} \)
79 \( 1 + 0.868iT - 79T^{2} \)
83 \( 1 + (-5.47 - 2.26i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-4.82 + 4.82i)T - 89iT^{2} \)
97 \( 1 - 6.97T + 97T^{2} \)
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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

−9.924140746761109836360632682992, −9.490067364902639931651991228012, −8.686343744890373448229254385550, −7.67507517119378773201312855474, −6.44147659186852987184025184761, −5.55834378218282918486254254202, −4.49464757480749582216648228059, −3.51495134129966830232421690861, −2.58393193780391529928689598022, −1.15137686135313693654285325425, 0.870591846606281094412837071044, 3.20917524436778083452997391521, 3.96128967052048070026697666361, 5.04918687163221341798311425121, 6.14040352914245157269068916435, 6.62603503514177526546875990368, 7.70254873295756158359024779357, 8.512273344178202212957734224290, 9.046206822528540264260522151206, 10.30163866538347090083555319937

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