Properties

Label 2-384-384.155-c1-0-57
Degree $2$
Conductor $384$
Sign $-0.579 + 0.814i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.862 − 1.12i)2-s + (0.969 − 1.43i)3-s + (−0.513 − 1.93i)4-s + (0.208 − 0.171i)5-s + (−0.773 − 2.32i)6-s + (0.829 + 1.24i)7-s + (−2.60 − 1.09i)8-s + (−1.12 − 2.78i)9-s + (−0.0120 − 0.381i)10-s + (2.00 − 0.607i)11-s + (−3.27 − 1.13i)12-s + (0.322 + 0.265i)13-s + (2.10 + 0.140i)14-s + (−0.0435 − 0.465i)15-s + (−3.47 + 1.98i)16-s + (−0.405 − 0.978i)17-s + ⋯
L(s)  = 1  + (0.609 − 0.792i)2-s + (0.559 − 0.828i)3-s + (−0.256 − 0.966i)4-s + (0.0932 − 0.0765i)5-s + (−0.315 − 0.948i)6-s + (0.313 + 0.468i)7-s + (−0.922 − 0.385i)8-s + (−0.373 − 0.927i)9-s + (−0.00380 − 0.120i)10-s + (0.604 − 0.183i)11-s + (−0.944 − 0.328i)12-s + (0.0895 + 0.0735i)13-s + (0.562 + 0.0375i)14-s + (−0.0112 − 0.120i)15-s + (−0.868 + 0.495i)16-s + (−0.0983 − 0.237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.579 + 0.814i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.579 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985849 - 1.91160i\)
\(L(\frac12)\) \(\approx\) \(0.985849 - 1.91160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.862 + 1.12i)T \)
3 \( 1 + (-0.969 + 1.43i)T \)
good5 \( 1 + (-0.208 + 0.171i)T + (0.975 - 4.90i)T^{2} \)
7 \( 1 + (-0.829 - 1.24i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (-2.00 + 0.607i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-0.322 - 0.265i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (0.405 + 0.978i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (3.56 - 0.350i)T + (18.6 - 3.70i)T^{2} \)
23 \( 1 + (-3.45 + 0.687i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-8.60 - 2.61i)T + (24.1 + 16.1i)T^{2} \)
31 \( 1 + (4.78 - 4.78i)T - 31iT^{2} \)
37 \( 1 + (-4.99 - 0.492i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (0.493 - 0.0981i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (0.863 - 1.61i)T + (-23.8 - 35.7i)T^{2} \)
47 \( 1 + (1.09 + 2.63i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-4.20 + 1.27i)T + (44.0 - 29.4i)T^{2} \)
59 \( 1 + (-7.56 + 6.21i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-4.21 - 7.88i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (4.34 - 2.32i)T + (37.2 - 55.7i)T^{2} \)
71 \( 1 + (3.44 + 5.15i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-2.06 - 1.37i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (11.1 + 4.61i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (4.64 - 0.457i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-1.76 + 8.87i)T + (-82.2 - 34.0i)T^{2} \)
97 \( 1 + (0.864 + 0.864i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.36977523000148532294796905831, −10.22657007226451572338923257618, −9.025049617182484619978163012342, −8.597520014653835856321972448235, −7.06278865983237970807858185818, −6.18356766965425173906756387974, −5.03615202418161081783268060276, −3.67392489857955907928072910318, −2.53704100095213474665822895021, −1.31995713269633353046823851749, 2.60989233511131506368189092585, 3.99822829088425803682408415087, 4.55196620489746179703491495110, 5.82645624183447362283027689458, 6.90607364506653563810468814334, 8.000162069138377595566356566201, 8.704838811355574104374833011054, 9.682846938910251108931165605716, 10.72346907327387656172409992619, 11.66012675252551144500324536773

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