Properties

Label 2-384-128.101-c1-0-26
Degree $2$
Conductor $384$
Sign $-0.833 + 0.551i$
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  + (−1.36 + 0.361i)2-s + (0.881 − 0.471i)3-s + (1.73 − 0.989i)4-s + (−2.41 − 2.94i)5-s + (−1.03 + 0.963i)6-s + (−0.0684 + 0.0457i)7-s + (−2.01 + 1.98i)8-s + (0.555 − 0.831i)9-s + (4.37 + 3.15i)10-s + (0.433 − 0.131i)11-s + (1.06 − 1.69i)12-s + (−2.71 − 2.22i)13-s + (0.0769 − 0.0872i)14-s + (−3.52 − 1.45i)15-s + (2.04 − 3.43i)16-s + (−6.00 + 2.48i)17-s + ⋯
L(s)  = 1  + (−0.966 + 0.255i)2-s + (0.509 − 0.272i)3-s + (0.868 − 0.494i)4-s + (−1.08 − 1.31i)5-s + (−0.422 + 0.393i)6-s + (−0.0258 + 0.0172i)7-s + (−0.713 + 0.700i)8-s + (0.185 − 0.277i)9-s + (1.38 + 0.997i)10-s + (0.130 − 0.0396i)11-s + (0.307 − 0.488i)12-s + (−0.752 − 0.617i)13-s + (0.0205 − 0.0233i)14-s + (−0.909 − 0.376i)15-s + (0.510 − 0.859i)16-s + (−1.45 + 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.833 + 0.551i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.833 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138330 - 0.459755i\)
\(L(\frac12)\) \(\approx\) \(0.138330 - 0.459755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.361i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (2.41 + 2.94i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (0.0684 - 0.0457i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-0.433 + 0.131i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (2.71 + 2.22i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (6.00 - 2.48i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.162 - 1.65i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (1.77 - 0.353i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.51 + 5.00i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (3.65 + 3.65i)T + 31iT^{2} \)
37 \( 1 + (-4.62 - 0.455i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (1.27 + 6.38i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.58 + 1.91i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (3.11 + 7.52i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-1.05 - 3.48i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-10.5 + 8.67i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (3.48 + 6.52i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (7.45 + 13.9i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-4.59 - 6.87i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-12.1 - 8.10i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-2.30 + 5.56i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-13.3 + 1.31i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-5.29 - 1.05i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (4.02 + 4.02i)T + 97iT^{2} \)
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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

−10.93789454695264418212166771640, −9.719070641937497839356603419883, −8.942527528615376835177540468723, −8.143244063742243186174333663815, −7.69565350186894454857024023495, −6.47412162638839353253383078761, −5.10961965780414660731054414585, −3.85312728074426805905884801980, −2.07794240521296366061902538178, −0.38622625893522191300278138872, 2.35945900088645533057799038745, 3.28043916263963767340516798005, 4.44936477928721611530045356296, 6.68599901557694680931269505964, 7.09239111027128160832683497323, 8.051285959063524539731436667263, 8.994921867727169544528056385062, 9.879097296635171861113608191701, 10.84032462973289734858143717649, 11.37788375500131795219107405360

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