Properties

Label 2-384-128.29-c1-0-11
Degree $2$
Conductor $384$
Sign $0.296 - 0.955i$
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  + (1.37 − 0.323i)2-s + (−0.0980 + 0.995i)3-s + (1.79 − 0.889i)4-s + (−3.05 + 1.63i)5-s + (0.186 + 1.40i)6-s + (0.897 + 4.51i)7-s + (2.17 − 1.80i)8-s + (−0.980 − 0.195i)9-s + (−3.68 + 3.23i)10-s + (−1.49 + 1.82i)11-s + (0.710 + 1.86i)12-s + (2.64 − 4.95i)13-s + (2.69 + 5.92i)14-s + (−1.32 − 3.20i)15-s + (2.41 − 3.18i)16-s + (−2.71 + 6.56i)17-s + ⋯
L(s)  = 1  + (0.973 − 0.228i)2-s + (−0.0565 + 0.574i)3-s + (0.895 − 0.444i)4-s + (−1.36 + 0.731i)5-s + (0.0762 + 0.572i)6-s + (0.339 + 1.70i)7-s + (0.770 − 0.637i)8-s + (−0.326 − 0.0650i)9-s + (−1.16 + 1.02i)10-s + (−0.450 + 0.549i)11-s + (0.204 + 0.539i)12-s + (0.734 − 1.37i)13-s + (0.719 + 1.58i)14-s + (−0.342 − 0.827i)15-s + (0.604 − 0.796i)16-s + (−0.659 + 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.296 - 0.955i$
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: \((2,\ 384,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56348 + 1.15222i\)
\(L(\frac12)\) \(\approx\) \(1.56348 + 1.15222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.323i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (3.05 - 1.63i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (-0.897 - 4.51i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (1.49 - 1.82i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (-2.64 + 4.95i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (2.71 - 6.56i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.494 - 1.62i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (-4.60 + 3.07i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-2.72 + 2.23i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (-2.98 + 2.98i)T - 31iT^{2} \)
37 \( 1 + (-2.73 - 0.828i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (-1.02 - 1.52i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.649 + 6.59i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (-3.79 - 1.57i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (7.75 + 6.36i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (1.13 + 2.11i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (7.50 + 0.738i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (-10.1 - 0.998i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (-10.0 + 2.00i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (2.68 - 13.5i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (6.59 - 2.72i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.58 - 0.481i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (3.98 + 2.66i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-8.26 + 8.26i)T - 97iT^{2} \)
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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

−11.44542655586911100047965783677, −10.94973314783374201500928300706, −10.11974242670732159140820762571, −8.507302375404874258611476791727, −7.908251329757045351261332290878, −6.45057408087042699406416489658, −5.58623753578970800662025751705, −4.51225908033464661657400238430, −3.43374710554650397716189485311, −2.49881810172648729309585508638, 1.04553835622531324306969207027, 3.23002878952405058576679046823, 4.34731049292259410840335491070, 4.88164088537035682907257211000, 6.63804615238666569344291188040, 7.29168759700282731991831535624, 7.943210717949410374349016147924, 9.014284704239571184478199730027, 10.96902371993251067731220556754, 11.25837116717708126700263957763

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