Properties

Label 2-384-8.3-c8-0-29
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $156.433$
Root an. cond. $12.5073$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.7·3-s + 197. i·5-s − 3.79e3i·7-s + 2.18e3·9-s + 1.64e4·11-s + 1.11e4i·13-s − 9.22e3i·15-s + 6.67e4·17-s + 2.04e4·19-s + 1.77e5i·21-s − 4.45e4i·23-s + 3.51e5·25-s − 1.02e5·27-s + 7.25e5i·29-s − 4.42e4i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.315i·5-s − 1.58i·7-s + 0.333·9-s + 1.12·11-s + 0.389i·13-s − 0.182i·15-s + 0.799·17-s + 0.157·19-s + 0.912i·21-s − 0.159i·23-s + 0.900·25-s − 0.192·27-s + 1.02i·29-s − 0.0478i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(156.433\)
Root analytic conductor: \(12.5073\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.079330301\)
\(L(\frac12)\) \(\approx\) \(2.079330301\)
\(L(5)\) 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 \( 1 + 46.7T \)
good5 \( 1 - 197. iT - 3.90e5T^{2} \)
7 \( 1 + 3.79e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.64e4T + 2.14e8T^{2} \)
13 \( 1 - 1.11e4iT - 8.15e8T^{2} \)
17 \( 1 - 6.67e4T + 6.97e9T^{2} \)
19 \( 1 - 2.04e4T + 1.69e10T^{2} \)
23 \( 1 + 4.45e4iT - 7.83e10T^{2} \)
29 \( 1 - 7.25e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.42e4iT - 8.52e11T^{2} \)
37 \( 1 - 2.85e6iT - 3.51e12T^{2} \)
41 \( 1 + 1.07e4T + 7.98e12T^{2} \)
43 \( 1 + 3.74e5T + 1.16e13T^{2} \)
47 \( 1 + 2.11e6iT - 2.38e13T^{2} \)
53 \( 1 - 5.99e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.11e7T + 1.46e14T^{2} \)
61 \( 1 + 2.30e7iT - 1.91e14T^{2} \)
67 \( 1 + 1.77e7T + 4.06e14T^{2} \)
71 \( 1 + 2.44e7iT - 6.45e14T^{2} \)
73 \( 1 - 3.70e6T + 8.06e14T^{2} \)
79 \( 1 - 5.98e7iT - 1.51e15T^{2} \)
83 \( 1 - 5.83e7T + 2.25e15T^{2} \)
89 \( 1 + 8.20e7T + 3.93e15T^{2} \)
97 \( 1 + 5.81e7T + 7.83e15T^{2} \)
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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

−10.17709470898445548864723959599, −9.246745568474937774621208037945, −7.950320407952445225677862617901, −6.91493639990728100229703601198, −6.53844220220356920755161599569, −5.08751469308049101494382761675, −4.12073680583371584655034493286, −3.27376713853987659802175948250, −1.44970219410906193815149110960, −0.78219401712516889175959458582, 0.63132826843217930861921557754, 1.74483705321450699127825914073, 2.95816161807724298190562531080, 4.25356794485866136443455590079, 5.46005156576759082988593841348, 5.92282921211737751388387951756, 7.06333406900329951361709060876, 8.304082580416301071148639112922, 9.099665102666480115474076139796, 9.841825521046919664160982507791

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