Properties

Label 2-384-128.29-c1-0-10
Degree $2$
Conductor $384$
Sign $-0.764 - 0.644i$
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  + (1.24 + 0.677i)2-s + (−0.0980 + 0.995i)3-s + (1.08 + 1.68i)4-s + (−2.78 + 1.48i)5-s + (−0.796 + 1.16i)6-s + (−0.0876 − 0.440i)7-s + (0.201 + 2.82i)8-s + (−0.980 − 0.195i)9-s + (−4.46 − 0.0397i)10-s + (1.26 − 1.53i)11-s + (−1.78 + 0.911i)12-s + (−2.72 + 5.09i)13-s + (0.189 − 0.606i)14-s + (−1.20 − 2.91i)15-s + (−1.66 + 3.63i)16-s + (2.38 − 5.74i)17-s + ⋯
L(s)  = 1  + (0.877 + 0.479i)2-s + (−0.0565 + 0.574i)3-s + (0.540 + 0.841i)4-s + (−1.24 + 0.666i)5-s + (−0.325 + 0.477i)6-s + (−0.0331 − 0.166i)7-s + (0.0714 + 0.997i)8-s + (−0.326 − 0.0650i)9-s + (−1.41 − 0.0125i)10-s + (0.380 − 0.464i)11-s + (−0.513 + 0.263i)12-s + (−0.755 + 1.41i)13-s + (0.0507 − 0.162i)14-s + (−0.312 − 0.753i)15-s + (−0.415 + 0.909i)16-s + (0.577 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.764 - 0.644i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.764 - 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567304 + 1.55307i\)
\(L(\frac12)\) \(\approx\) \(0.567304 + 1.55307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.677i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (2.78 - 1.48i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.0876 + 0.440i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-1.26 + 1.53i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (2.72 - 5.09i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (-2.38 + 5.74i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.73 - 5.72i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (0.164 - 0.110i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-5.96 + 4.89i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (5.34 - 5.34i)T - 31iT^{2} \)
37 \( 1 + (-11.0 - 3.36i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (0.387 + 0.580i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.575 + 5.84i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (-11.4 - 4.74i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (3.96 + 3.25i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (-3.86 - 7.23i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (0.782 + 0.0770i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (5.42 + 0.534i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (5.54 - 1.10i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-0.466 + 2.34i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (1.00 - 0.418i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (2.85 - 0.864i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (8.58 + 5.73i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-11.3 + 11.3i)T - 97iT^{2} \)
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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

−11.82945971787681692209095884556, −11.16506959167964514073478845712, −9.972099847909306666256905792967, −8.771350473304388841671123958098, −7.60062678303586991729868218960, −7.07340622356412092230763805036, −5.88557758423168298057165358160, −4.58246264677802183091519663496, −3.86492771033964453808547027252, −2.85678575226985753321875441899, 0.877111624572503331399463385260, 2.69704968787260358947493958238, 3.94193663183463944716043046423, 4.94106494074250743855106962271, 5.95692757897655515833713800170, 7.27177387220187536197903814482, 7.939160917113142780222803577010, 9.170449030482260045892360913208, 10.39535376496742466227862506022, 11.29610027420274649807653182247

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