Properties

Label 2-384-128.13-c1-0-1
Degree $2$
Conductor $384$
Sign $-0.871 + 0.489i$
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  + (−0.870 + 1.11i)2-s + (0.290 + 0.956i)3-s + (−0.484 − 1.94i)4-s + (−0.328 − 3.33i)5-s + (−1.31 − 0.509i)6-s + (−1.53 + 2.29i)7-s + (2.58 + 1.15i)8-s + (−0.831 + 0.555i)9-s + (4.00 + 2.53i)10-s + (−2.66 + 1.42i)11-s + (1.71 − 1.02i)12-s + (−6.16 − 0.607i)13-s + (−1.22 − 3.70i)14-s + (3.09 − 1.28i)15-s + (−3.53 + 1.87i)16-s + (1.52 + 0.633i)17-s + ⋯
L(s)  = 1  + (−0.615 + 0.788i)2-s + (0.167 + 0.552i)3-s + (−0.242 − 0.970i)4-s + (−0.146 − 1.49i)5-s + (−0.538 − 0.208i)6-s + (−0.579 + 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.277 + 0.185i)9-s + (1.26 + 0.802i)10-s + (−0.803 + 0.429i)11-s + (0.495 − 0.296i)12-s + (−1.70 − 0.168i)13-s + (−0.326 − 0.989i)14-s + (0.799 − 0.331i)15-s + (−0.882 + 0.469i)16-s + (0.370 + 0.153i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.871 + 0.489i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.871 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0360648 - 0.137817i\)
\(L(\frac12)\) \(\approx\) \(0.0360648 - 0.137817i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.870 - 1.11i)T \)
3 \( 1 + (-0.290 - 0.956i)T \)
good5 \( 1 + (0.328 + 3.33i)T + (-4.90 + 0.975i)T^{2} \)
7 \( 1 + (1.53 - 2.29i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (2.66 - 1.42i)T + (6.11 - 9.14i)T^{2} \)
13 \( 1 + (6.16 + 0.607i)T + (12.7 + 2.53i)T^{2} \)
17 \( 1 + (-1.52 - 0.633i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (6.31 - 5.18i)T + (3.70 - 18.6i)T^{2} \)
23 \( 1 + (0.448 - 2.25i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.20 + 2.24i)T + (-16.1 - 24.1i)T^{2} \)
31 \( 1 + (-0.352 + 0.352i)T - 31iT^{2} \)
37 \( 1 + (-1.37 + 1.67i)T + (-7.21 - 36.2i)T^{2} \)
41 \( 1 + (-0.719 - 0.143i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.498 + 1.64i)T + (-35.7 - 23.8i)T^{2} \)
47 \( 1 + (-5.00 + 12.0i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (-1.75 - 3.28i)T + (-29.4 + 44.0i)T^{2} \)
59 \( 1 + (3.13 - 0.308i)T + (57.8 - 11.5i)T^{2} \)
61 \( 1 + (9.72 - 2.94i)T + (50.7 - 33.8i)T^{2} \)
67 \( 1 + (-1.59 + 0.484i)T + (55.7 - 37.2i)T^{2} \)
71 \( 1 + (11.9 + 7.97i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (2.78 + 4.16i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-5.23 - 12.6i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (0.835 + 1.01i)T + (-16.1 + 81.4i)T^{2} \)
89 \( 1 + (-3.31 - 16.6i)T + (-82.2 + 34.0i)T^{2} \)
97 \( 1 + (-3.58 + 3.58i)T - 97iT^{2} \)
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

−12.15987522709681753512181436315, −10.44624815188984628372251193099, −9.795621735716508282128817193038, −9.066734396897628180406116211632, −8.291086862866747304857894948439, −7.51332522378606291339225202019, −5.92688475847800764072969802171, −5.20736272664810772409989912973, −4.34949790943518841379824393063, −2.20640030481755034035819807759, 0.10407415054866883791787890786, 2.49057663661399946279777840675, 3.03812106394253450483133435401, 4.48517031145122238252388701270, 6.51639648193960193598126843529, 7.22713897788024614043562672894, 7.81786435819478119715401406632, 9.137844859464752770868880039115, 10.26195037681339935084383777292, 10.59837516679449440468517723239

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