Properties

Label 2-384-48.11-c1-0-0
Degree $2$
Conductor $384$
Sign $0.0993 - 0.995i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
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 − 0.0835i)3-s + (0.431 − 0.431i)5-s − 3.10·7-s + (2.98 + 0.289i)9-s + (2.98 + 2.98i)11-s + (−2.10 + 2.10i)13-s + (−0.782 + 0.710i)15-s + 2.42i·17-s + (0.710 + 0.710i)19-s + (5.36 + 0.259i)21-s + 5.97i·23-s + 4.62i·25-s + (−5.14 − 0.749i)27-s + (2.86 + 2.86i)29-s − 0.524i·31-s + ⋯
L(s)  = 1  + (−0.998 − 0.0482i)3-s + (0.193 − 0.193i)5-s − 1.17·7-s + (0.995 + 0.0963i)9-s + (0.900 + 0.900i)11-s + (−0.583 + 0.583i)13-s + (−0.202 + 0.183i)15-s + 0.589i·17-s + (0.163 + 0.163i)19-s + (1.17 + 0.0565i)21-s + 1.24i·23-s + 0.925i·25-s + (−0.989 − 0.144i)27-s + (0.531 + 0.531i)29-s − 0.0941i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0993 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0993 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.0993 - 0.995i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.0993 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534895 + 0.484166i\)
\(L(\frac12)\) \(\approx\) \(0.534895 + 0.484166i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0835i)T \)
good5 \( 1 + (-0.431 + 0.431i)T - 5iT^{2} \)
7 \( 1 + 3.10T + 7T^{2} \)
11 \( 1 + (-2.98 - 2.98i)T + 11iT^{2} \)
13 \( 1 + (2.10 - 2.10i)T - 13iT^{2} \)
17 \( 1 - 2.42iT - 17T^{2} \)
19 \( 1 + (-0.710 - 0.710i)T + 19iT^{2} \)
23 \( 1 - 5.97iT - 23T^{2} \)
29 \( 1 + (-2.86 - 2.86i)T + 29iT^{2} \)
31 \( 1 + 0.524iT - 31T^{2} \)
37 \( 1 + (1.52 + 1.52i)T + 37iT^{2} \)
41 \( 1 + 1.81T + 41T^{2} \)
43 \( 1 + (0.710 - 0.710i)T - 43iT^{2} \)
47 \( 1 - 7.53T + 47T^{2} \)
53 \( 1 + (8.83 - 8.83i)T - 53iT^{2} \)
59 \( 1 + (0.0804 + 0.0804i)T + 59iT^{2} \)
61 \( 1 + (-5.72 + 5.72i)T - 61iT^{2} \)
67 \( 1 + (-0.391 - 0.391i)T + 67iT^{2} \)
71 \( 1 + 5.01iT - 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 + 3.47iT - 79T^{2} \)
83 \( 1 + (-4.55 + 4.55i)T - 83iT^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + 8.67T + 97T^{2} \)
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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

−11.71734964525765840277776219136, −10.61432293722322195779970941879, −9.649120871106100111504567473672, −9.254319044774285272126415074659, −7.45646727919855533188772288679, −6.73489497686664171795110275033, −5.89245370937559322197688343165, −4.75421917768857325426330451083, −3.61188680304854987620891507409, −1.62264120640177571896522724276, 0.56065310157211185376996434404, 2.84114913044309593773789383027, 4.17097972430391015050659198082, 5.45757748409942254461052348513, 6.41700497918869511562420813634, 6.91203165607732151414413220438, 8.407190597383261080358880345323, 9.618143283009180310638533229704, 10.17659653543097258925556995423, 11.13415107203054131644526794436

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