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Properties

Label 2-384-8.3-c2-0-6
Degree $2$
Conductor $384$
Sign $0.707 - 0.707i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s + 0.898i·5-s + 2.82i·7-s + 2.99·9-s + 4.38·11-s + 13.7i·13-s + 1.55i·15-s + 17.5·17-s − 4.38·19-s + 4.89i·21-s + 22.0i·23-s + 24.1·25-s + 5.19·27-s + 44.4i·29-s − 53.1i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.179i·5-s + 0.404i·7-s + 0.333·9-s + 0.398·11-s + 1.06i·13-s + 0.103i·15-s + 1.03·17-s − 0.230·19-s + 0.233i·21-s + 0.958i·23-s + 0.967·25-s + 0.192·27-s + 1.53i·29-s − 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.94437 + 0.805386i\)
\(L(\frac12)\) \(\approx\) \(1.94437 + 0.805386i\)
\(L(2)\) 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 \( 1 - 1.73T \)
good5 \( 1 - 0.898iT - 25T^{2} \)
7 \( 1 - 2.82iT - 49T^{2} \)
11 \( 1 - 4.38T + 121T^{2} \)
13 \( 1 - 13.7iT - 169T^{2} \)
17 \( 1 - 17.5T + 289T^{2} \)
19 \( 1 + 4.38T + 361T^{2} \)
23 \( 1 - 22.0iT - 529T^{2} \)
29 \( 1 - 44.4iT - 841T^{2} \)
31 \( 1 + 53.1iT - 961T^{2} \)
37 \( 1 - 35.1iT - 1.36e3T^{2} \)
41 \( 1 - 37.5T + 1.68e3T^{2} \)
43 \( 1 + 49.6T + 1.84e3T^{2} \)
47 \( 1 + 38.4iT - 2.20e3T^{2} \)
53 \( 1 + 1.70iT - 2.80e3T^{2} \)
59 \( 1 - 34.6T + 3.48e3T^{2} \)
61 \( 1 + 24.4iT - 3.72e3T^{2} \)
67 \( 1 + 93.7T + 4.48e3T^{2} \)
71 \( 1 - 123. iT - 5.04e3T^{2} \)
73 \( 1 + 10T + 5.32e3T^{2} \)
79 \( 1 + 131. iT - 6.24e3T^{2} \)
83 \( 1 - 110.T + 6.88e3T^{2} \)
89 \( 1 + 73.1T + 7.92e3T^{2} \)
97 \( 1 + 105.T + 9.40e3T^{2} \)
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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

−11.34909881943724613388055673417, −10.17129418913650425981160437013, −9.321707614761428162538304753609, −8.599774546783128876699540154102, −7.49631805892605710029205197048, −6.61265431177960003481889819111, −5.40231636281206399244207117109, −4.10892874441146498788447043225, −2.98666730153954462294220496686, −1.58005507792194573823468831628, 0.986987713126649011712094129468, 2.72498663524746244016189514844, 3.85158081887047015371004729627, 5.04042996098395006778367153140, 6.27964726279257400065906697744, 7.41875543435543240176239391153, 8.225395544302396386870417150214, 9.103469699898379597851136897941, 10.16576564020355425176624720161, 10.76115577921119155618535327300

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