Properties

Label 2-864-32.5-c1-0-18
Degree $2$
Conductor $864$
Sign $0.0631 - 0.998i$
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  + (−1.12 + 0.861i)2-s + (0.515 − 1.93i)4-s + (1.04 + 0.433i)5-s + (2.00 + 2.00i)7-s + (1.08 + 2.61i)8-s + (−1.54 + 0.414i)10-s + (−0.338 + 0.818i)11-s + (1.78 − 0.738i)13-s + (−3.98 − 0.522i)14-s + (−3.46 − 1.99i)16-s + 5.63i·17-s + (4.50 − 1.86i)19-s + (1.37 − 1.79i)20-s + (−0.324 − 1.20i)22-s + (−3.31 + 3.31i)23-s + ⋯
L(s)  = 1  + (−0.793 + 0.609i)2-s + (0.257 − 0.966i)4-s + (0.467 + 0.193i)5-s + (0.758 + 0.758i)7-s + (0.383 + 0.923i)8-s + (−0.488 + 0.131i)10-s + (−0.102 + 0.246i)11-s + (0.494 − 0.204i)13-s + (−1.06 − 0.139i)14-s + (−0.866 − 0.498i)16-s + 1.36i·17-s + (1.03 − 0.427i)19-s + (0.307 − 0.401i)20-s + (−0.0692 − 0.257i)22-s + (−0.690 + 0.690i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0631 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.0631 - 0.998i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.0631 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902734 + 0.847386i\)
\(L(\frac12)\) \(\approx\) \(0.902734 + 0.847386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.861i)T \)
3 \( 1 \)
good5 \( 1 + (-1.04 - 0.433i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.00 - 2.00i)T + 7iT^{2} \)
11 \( 1 + (0.338 - 0.818i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.78 + 0.738i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 5.63iT - 17T^{2} \)
19 \( 1 + (-4.50 + 1.86i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.31 - 3.31i)T - 23iT^{2} \)
29 \( 1 + (0.803 + 1.93i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 2.71T + 31T^{2} \)
37 \( 1 + (-5.51 - 2.28i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-7.12 + 7.12i)T - 41iT^{2} \)
43 \( 1 + (1.98 - 4.79i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 4.34iT - 47T^{2} \)
53 \( 1 + (0.634 - 1.53i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.37 - 1.39i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-4.15 - 10.0i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-4.91 - 11.8i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-5.53 - 5.53i)T + 71iT^{2} \)
73 \( 1 + (-1.43 + 1.43i)T - 73iT^{2} \)
79 \( 1 + 3.15iT - 79T^{2} \)
83 \( 1 + (14.6 - 6.08i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.54 + 8.54i)T + 89iT^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−10.10715104811688659489856375270, −9.524477630637524229270285384740, −8.535328366996413898138140129249, −7.997853117301886785081444323731, −7.06848799736936797595566717925, −5.79925666787301516336127955030, −5.66731127280205800421488132607, −4.23383288622320143182613367012, −2.47306440012043137090677399605, −1.44582765244647713519370431961, 0.867316200097242532593017368994, 2.01915234065903684502139567606, 3.33964798285234138617880875255, 4.39115480684505330980059372129, 5.51766321316503346027775432401, 6.80279937289850562005720600305, 7.65913022892994383582060726646, 8.274391615212877246350037932237, 9.406297415906321664813867958278, 9.776830069771337968596695937096

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