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Properties

Label 2-384-8.5-c7-0-38
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 27i·3-s + 69.8i·5-s + 860.·7-s − 729·9-s − 6.10e3i·11-s + 5.33e3i·13-s + 1.88e3·15-s + 3.86e4·17-s + 1.35e4i·19-s − 2.32e4i·21-s − 9.96e4·23-s + 7.32e4·25-s + 1.96e4i·27-s + 8.11e4i·29-s + 1.53e5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.250i·5-s + 0.947·7-s − 0.333·9-s − 1.38i·11-s + 0.673i·13-s + 0.144·15-s + 1.90·17-s + 0.454i·19-s − 0.547i·21-s − 1.70·23-s + 0.937·25-s + 0.192i·27-s + 0.618i·29-s + 0.922·31-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

\(L(4)\) \(\approx\) \(2.696736284\)
\(L(\frac12)\) \(\approx\) \(2.696736284\)
\(L(\frac{9}{2})\) 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 \( 1 - 69.8iT - 7.81e4T^{2} \)
7 \( 1 - 860.T + 8.23e5T^{2} \)
11 \( 1 + 6.10e3iT - 1.94e7T^{2} \)
13 \( 1 - 5.33e3iT - 6.27e7T^{2} \)
17 \( 1 - 3.86e4T + 4.10e8T^{2} \)
19 \( 1 - 1.35e4iT - 8.93e8T^{2} \)
23 \( 1 + 9.96e4T + 3.40e9T^{2} \)
29 \( 1 - 8.11e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.53e5T + 2.75e10T^{2} \)
37 \( 1 + 4.67e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.84e5T + 1.94e11T^{2} \)
43 \( 1 + 4.50e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.03e5T + 5.06e11T^{2} \)
53 \( 1 - 1.61e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.58e6iT - 2.48e12T^{2} \)
61 \( 1 - 2.61e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.52e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.15e6T + 9.09e12T^{2} \)
73 \( 1 - 3.52e6T + 1.10e13T^{2} \)
79 \( 1 - 7.64e6T + 1.92e13T^{2} \)
83 \( 1 + 6.34e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.70e6T + 4.42e13T^{2} \)
97 \( 1 + 2.70e6T + 8.07e13T^{2} \)
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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

−10.19571086819454050468808247693, −8.898386358805547763563205642280, −8.087969027707307793089826720752, −7.41945935581569372382415461920, −6.12118202339675846420240115800, −5.47472098873521621220246489279, −4.04154485525338483877717246423, −2.92366085247958399269290819395, −1.63867205317007382070271855141, −0.73458187609075718150336185087, 0.872899795729505294207254685799, 2.05269452421564668918993536401, 3.37885786435595855973513588789, 4.64218575670161950032730126519, 5.13846652632572681810687183172, 6.40296012243668294851028809894, 7.893209923073739030105107327298, 8.124912646319523804036413480408, 9.761489935590850949030931508977, 9.952371391439779339866248391988

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