Properties

Label 2-384-8.5-c7-0-26
Degree $2$
Conductor $384$
Sign $-0.707 - 0.707i$
Analytic cond. $119.955$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 27i·3-s + 522. i·5-s + 1.30e3·7-s − 729·9-s − 2.79e3i·11-s − 7.35e3i·13-s − 1.41e4·15-s + 5.64e3·17-s + 2.02e4i·19-s + 3.53e4i·21-s + 7.95e4·23-s − 1.94e5·25-s − 1.96e4i·27-s + 1.40e5i·29-s + 3.21e5·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.86i·5-s + 1.44·7-s − 0.333·9-s − 0.633i·11-s − 0.928i·13-s − 1.07·15-s + 0.278·17-s + 0.678i·19-s + 0.832i·21-s + 1.36·23-s − 2.49·25-s − 0.192i·27-s + 1.06i·29-s + 1.93·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(119.955\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.728356624\)
\(L(\frac12)\) \(\approx\) \(2.728356624\)
\(L(\frac{9}{2})\) 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 \( 1 - 27iT \)
good5 \( 1 - 522. iT - 7.81e4T^{2} \)
7 \( 1 - 1.30e3T + 8.23e5T^{2} \)
11 \( 1 + 2.79e3iT - 1.94e7T^{2} \)
13 \( 1 + 7.35e3iT - 6.27e7T^{2} \)
17 \( 1 - 5.64e3T + 4.10e8T^{2} \)
19 \( 1 - 2.02e4iT - 8.93e8T^{2} \)
23 \( 1 - 7.95e4T + 3.40e9T^{2} \)
29 \( 1 - 1.40e5iT - 1.72e10T^{2} \)
31 \( 1 - 3.21e5T + 2.75e10T^{2} \)
37 \( 1 - 5.04e5iT - 9.49e10T^{2} \)
41 \( 1 - 2.62e5T + 1.94e11T^{2} \)
43 \( 1 - 5.03e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.07e5T + 5.06e11T^{2} \)
53 \( 1 + 1.13e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.80e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.00e6iT - 3.14e12T^{2} \)
67 \( 1 - 4.70e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.38e6T + 9.09e12T^{2} \)
73 \( 1 + 1.51e6T + 1.10e13T^{2} \)
79 \( 1 - 1.44e5T + 1.92e13T^{2} \)
83 \( 1 + 1.45e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.58e6T + 4.42e13T^{2} \)
97 \( 1 + 8.08e6T + 8.07e13T^{2} \)
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

−10.55195548420356931210483396984, −9.966166113024904776492393328473, −8.447809078151306837572183676239, −7.81213177615936337444533423494, −6.74076941955920721800887988430, −5.72154840969156734087202111170, −4.68941149912498581534449410011, −3.32458043925607657050604824846, −2.73887222241377493490262531550, −1.20807926282437696209435889529, 0.61949442224616038518966907070, 1.36557729240289961601153599207, 2.23274680215755362895878004662, 4.39627920907412757643615712616, 4.75573130994035851435428890852, 5.78552824083680248787098081253, 7.23310366707401480944225715316, 8.047982726367541383734119271748, 8.815792755948344889874797617930, 9.451897760301195365070510263024

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