↑

Properties

Label 2-384-16.11-c6-0-19
Degree $2$
Conductor $384$
Sign $0.203 + 0.979i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−11.0 − 11.0i)3-s + (9.26 + 9.26i)5-s − 320.·7-s + 242. i·9-s + (−1.49e3 + 1.49e3i)11-s + (−2.54e3 + 2.54e3i)13-s − 204. i·15-s − 7.23e3·17-s + (4.86e3 + 4.86e3i)19-s + (3.53e3 + 3.53e3i)21-s + 1.22e4·23-s − 1.54e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−9.52e3 + 9.52e3i)29-s − 1.35e4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.0741 + 0.0741i)5-s − 0.934·7-s + 0.333i·9-s + (−1.12 + 1.12i)11-s + (−1.15 + 1.15i)13-s − 0.0605i·15-s − 1.47·17-s + (0.709 + 0.709i)19-s + (0.381 + 0.381i)21-s + 1.00·23-s − 0.989i·25-s + (0.136 − 0.136i)27-s + (−0.390 + 0.390i)29-s − 0.453i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ 0.203 + 0.979i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.2886009913\)
\(L(\frac12)\) \(\approx\) \(0.2886009913\)
\(L(4)\) 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 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (-9.26 - 9.26i)T + 1.56e4iT^{2} \)
7 \( 1 + 320.T + 1.17e5T^{2} \)
11 \( 1 + (1.49e3 - 1.49e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (2.54e3 - 2.54e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 7.23e3T + 2.41e7T^{2} \)
19 \( 1 + (-4.86e3 - 4.86e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.22e4T + 1.48e8T^{2} \)
29 \( 1 + (9.52e3 - 9.52e3i)T - 5.94e8iT^{2} \)
31 \( 1 + 1.35e4iT - 8.87e8T^{2} \)
37 \( 1 + (-964. - 964. i)T + 2.56e9iT^{2} \)
41 \( 1 - 9.46e4iT - 4.75e9T^{2} \)
43 \( 1 + (6.14e4 - 6.14e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.25e5iT - 1.07e10T^{2} \)
53 \( 1 + (9.27e4 + 9.27e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (-7.89e4 + 7.89e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.40e4 - 1.40e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (-1.69e5 - 1.69e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.39e5T + 1.28e11T^{2} \)
73 \( 1 - 6.68e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.72e5iT - 2.43e11T^{2} \)
83 \( 1 + (-3.60e5 - 3.60e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.01e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.29e6T + 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.945155184777533593657981503259, −9.535412937310831456488452000269, −8.147425804411430497500362739266, −7.00940699017651254827797535390, −6.64537831185437288226102202038, −5.23121280622825129534756782853, −4.40246488972436119233073689933, −2.76528596395414415375194518554, −1.88308566087674496006699681816, −0.11862933747041119273279469195, 0.55425421630593587647862333947, 2.62748691288809041382157749220, 3.36870665936006359036954930344, 4.95430975325799938560011795786, 5.54114933970199916261281737205, 6.71217837900261541449507448026, 7.66820111393422979071040495198, 8.906231494841261739740226783586, 9.630452373066918994257228077253, 10.66732604687883322134455603926

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