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Properties

Label 2-864-32.13-c1-0-49
Degree $2$
Conductor $864$
Sign $-0.0584 + 0.998i$
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  + (−1.18 + 0.771i)2-s + (0.808 − 1.82i)4-s + (−0.796 + 0.329i)5-s + (3.41 − 3.41i)7-s + (0.454 + 2.79i)8-s + (0.689 − 1.00i)10-s + (−0.811 − 1.95i)11-s + (−3.44 − 1.42i)13-s + (−1.41 + 6.68i)14-s + (−2.69 − 2.95i)16-s − 5.12i·17-s + (1.84 + 0.763i)19-s + (−0.0403 + 1.72i)20-s + (2.47 + 1.69i)22-s + (−0.655 − 0.655i)23-s + ⋯
L(s)  = 1  + (−0.837 + 0.545i)2-s + (0.404 − 0.914i)4-s + (−0.356 + 0.147i)5-s + (1.29 − 1.29i)7-s + (0.160 + 0.987i)8-s + (0.217 − 0.318i)10-s + (−0.244 − 0.590i)11-s + (−0.956 − 0.396i)13-s + (−0.376 + 1.78i)14-s + (−0.673 − 0.739i)16-s − 1.24i·17-s + (0.423 + 0.175i)19-s + (−0.00902 + 0.385i)20-s + (0.527 + 0.361i)22-s + (−0.136 − 0.136i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0584 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0584 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.0584 + 0.998i$
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: \((2,\ 864,\ (\ :1/2),\ -0.0584 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505219 - 0.535671i\)
\(L(\frac12)\) \(\approx\) \(0.505219 - 0.535671i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.771i)T \)
3 \( 1 \)
good5 \( 1 + (0.796 - 0.329i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.41 + 3.41i)T - 7iT^{2} \)
11 \( 1 + (0.811 + 1.95i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (3.44 + 1.42i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.12iT - 17T^{2} \)
19 \( 1 + (-1.84 - 0.763i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.655 + 0.655i)T + 23iT^{2} \)
29 \( 1 + (3.09 - 7.48i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.46T + 31T^{2} \)
37 \( 1 + (-1.04 + 0.431i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.85 + 7.85i)T + 41iT^{2} \)
43 \( 1 + (-1.55 - 3.74i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 8.53iT - 47T^{2} \)
53 \( 1 + (-2.21 - 5.35i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.55 - 1.47i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.53 + 13.3i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.49 + 3.61i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-3.65 + 3.65i)T - 71iT^{2} \)
73 \( 1 + (1.36 + 1.36i)T + 73iT^{2} \)
79 \( 1 - 9.69iT - 79T^{2} \)
83 \( 1 + (1.86 + 0.772i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-9.11 + 9.11i)T - 89iT^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−9.943193160120492161914430650784, −9.019678353941983226213380644083, −8.040619688048811887430088377703, −7.39516090694266481278243353227, −7.05704575646569031389011496672, −5.42121479674594080329820703674, −4.92542233663109326209773797503, −3.49702497891745503736048653594, −1.85449087396535487879174016721, −0.45876241824356615269717765485, 1.76200120677950003023677312703, 2.43174879518359635500312267695, 3.99090181249589140329475379821, 4.94826510024056308607137542777, 6.07362596969285932084364988307, 7.44269031882010409294477503385, 7.977121110685215839845331705136, 8.703108114942655065321641123775, 9.514012427116857358291204772665, 10.29383812993139103899924532897

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