↑

Properties

Label 2-384-48.11-c3-0-34
Degree $2$
Conductor $384$
Sign $-0.517 + 0.855i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.16 − 0.563i)3-s + (−13.1 + 13.1i)5-s − 13.2·7-s + (26.3 − 5.82i)9-s + (−24.0 − 24.0i)11-s + (30.7 − 30.7i)13-s + (−60.5 + 75.4i)15-s + 56.8i·17-s + (−74.2 − 74.2i)19-s + (−68.5 + 7.47i)21-s − 21.7i·23-s − 221. i·25-s + (132. − 44.9i)27-s + (−102. − 102. i)29-s − 219. i·31-s + ⋯
L(s)  = 1  + (0.994 − 0.108i)3-s + (−1.17 + 1.17i)5-s − 0.716·7-s + (0.976 − 0.215i)9-s + (−0.660 − 0.660i)11-s + (0.656 − 0.656i)13-s + (−1.04 + 1.29i)15-s + 0.811i·17-s + (−0.896 − 0.896i)19-s + (−0.712 + 0.0777i)21-s − 0.197i·23-s − 1.77i·25-s + (0.947 − 0.320i)27-s + (−0.657 − 0.657i)29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.517 + 0.855i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.517 + 0.855i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7129653917\)
\(L(\frac12)\) \(\approx\) \(0.7129653917\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.16 + 0.563i)T \)
good5 \( 1 + (13.1 - 13.1i)T - 125iT^{2} \)
7 \( 1 + 13.2T + 343T^{2} \)
11 \( 1 + (24.0 + 24.0i)T + 1.33e3iT^{2} \)
13 \( 1 + (-30.7 + 30.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 56.8iT - 4.91e3T^{2} \)
19 \( 1 + (74.2 + 74.2i)T + 6.85e3iT^{2} \)
23 \( 1 + 21.7iT - 1.21e4T^{2} \)
29 \( 1 + (102. + 102. i)T + 2.43e4iT^{2} \)
31 \( 1 + 219. iT - 2.97e4T^{2} \)
37 \( 1 + (83.3 + 83.3i)T + 5.06e4iT^{2} \)
41 \( 1 + 7.92T + 6.89e4T^{2} \)
43 \( 1 + (-153. + 153. i)T - 7.95e4iT^{2} \)
47 \( 1 + 208.T + 1.03e5T^{2} \)
53 \( 1 + (390. - 390. i)T - 1.48e5iT^{2} \)
59 \( 1 + (221. + 221. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-416. + 416. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-284. - 284. i)T + 3.00e5iT^{2} \)
71 \( 1 - 26.9iT - 3.57e5T^{2} \)
73 \( 1 - 839. iT - 3.89e5T^{2} \)
79 \( 1 + 556. iT - 4.93e5T^{2} \)
83 \( 1 + (400. - 400. i)T - 5.71e5iT^{2} \)
89 \( 1 + 974.T + 7.04e5T^{2} \)
97 \( 1 + 1.55e3T + 9.12e5T^{2} \)
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

−10.73225188744395875554639432081, −9.723881257797269851214293971756, −8.465008481315646647064400299457, −7.926754761823601517681899550270, −6.97876185451965821814990709453, −6.07502438503125686225405363332, −4.10973911582852236183959357017, −3.36980816558809304313250792619, −2.51187740055850651915511359915, −0.20937071375927643282688423530, 1.56895306947396642675473413607, 3.23564639034158830227754239588, 4.15270620112957488034455656157, 5.03415385528870835500677871960, 6.76772057467236297651455128612, 7.72708866826936044710458321440, 8.489319536339174192167154789034, 9.187871398289878969685774538235, 10.08151709064829950650259507971, 11.27835872844644737564878836826

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