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Properties

Label 2-384-4.3-c8-0-2
Degree $2$
Conductor $384$
Sign $i$
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  − 46.7i·3-s − 1.08e3·5-s + 399. i·7-s − 2.18e3·9-s + 4.27e3i·11-s − 3.80e4·13-s + 5.06e4i·15-s − 1.25e5·17-s + 2.16e5i·19-s + 1.86e4·21-s + 3.00e5i·23-s + 7.83e5·25-s + 1.02e5i·27-s − 4.34e5·29-s − 7.87e5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.73·5-s + 0.166i·7-s − 0.333·9-s + 0.291i·11-s − 1.33·13-s + 1.00i·15-s − 1.49·17-s + 1.66i·19-s + 0.0959·21-s + 1.07i·23-s + 2.00·25-s + 0.192i·27-s − 0.613·29-s − 0.852i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(156.433\)
Root analytic conductor: \(12.5073\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :4),\ i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.09074248165\)
\(L(\frac12)\) \(\approx\) \(0.09074248165\)
\(L(5)\) 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 \( 1 + 46.7iT \)
good5 \( 1 + 1.08e3T + 3.90e5T^{2} \)
7 \( 1 - 399. iT - 5.76e6T^{2} \)
11 \( 1 - 4.27e3iT - 2.14e8T^{2} \)
13 \( 1 + 3.80e4T + 8.15e8T^{2} \)
17 \( 1 + 1.25e5T + 6.97e9T^{2} \)
19 \( 1 - 2.16e5iT - 1.69e10T^{2} \)
23 \( 1 - 3.00e5iT - 7.83e10T^{2} \)
29 \( 1 + 4.34e5T + 5.00e11T^{2} \)
31 \( 1 + 7.87e5iT - 8.52e11T^{2} \)
37 \( 1 + 1.35e6T + 3.51e12T^{2} \)
41 \( 1 + 6.23e5T + 7.98e12T^{2} \)
43 \( 1 - 4.76e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.12e6iT - 2.38e13T^{2} \)
53 \( 1 + 7.12e6T + 6.22e13T^{2} \)
59 \( 1 - 1.33e7iT - 1.46e14T^{2} \)
61 \( 1 + 2.19e7T + 1.91e14T^{2} \)
67 \( 1 - 1.18e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.35e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.79e7T + 8.06e14T^{2} \)
79 \( 1 - 3.52e7iT - 1.51e15T^{2} \)
83 \( 1 + 2.93e6iT - 2.25e15T^{2} \)
89 \( 1 + 9.53e7T + 3.93e15T^{2} \)
97 \( 1 - 7.13e7T + 7.83e15T^{2} \)
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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

−10.94048998968338608971884783527, −9.677120441332278631925456246627, −8.617845482582962606068516891766, −7.59970322678493949273092483610, −7.38742124607488848293181803528, −6.08119279231316183765359690146, −4.72908176041253533713527101101, −3.90912006260847558613116909795, −2.74724516276637316702674857221, −1.49910143871217065580334346776, 0.05741883374922585726856364123, 0.28986349261010313421516687800, 2.41078503678779361966764582577, 3.45324577499176972025499120340, 4.47609738856009533740919288810, 4.95906399879765985040671282023, 6.78748054353932463354778797214, 7.33826558832884125081588720914, 8.537065546127106452924765046592, 9.050058287590672306238971098751

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