↑

Properties

Label 2-384-128.45-c1-0-4
Degree $2$
Conductor $384$
Sign $0.999 + 0.0118i$
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  + (0.249 − 1.39i)2-s + (−0.471 + 0.881i)3-s + (−1.87 − 0.693i)4-s + (0.986 + 0.809i)5-s + (1.11 + 0.875i)6-s + (1.86 + 1.24i)7-s + (−1.43 + 2.43i)8-s + (−0.555 − 0.831i)9-s + (1.37 − 1.17i)10-s + (−0.801 + 2.64i)11-s + (1.49 − 1.32i)12-s + (0.812 + 0.989i)13-s + (2.20 − 2.29i)14-s + (−1.17 + 0.488i)15-s + (3.03 + 2.60i)16-s + (6.22 + 2.57i)17-s + ⋯
L(s)  = 1  + (0.176 − 0.984i)2-s + (−0.272 + 0.509i)3-s + (−0.937 − 0.346i)4-s + (0.441 + 0.361i)5-s + (0.453 + 0.357i)6-s + (0.706 + 0.472i)7-s + (−0.506 + 0.862i)8-s + (−0.185 − 0.277i)9-s + (0.434 − 0.370i)10-s + (−0.241 + 0.796i)11-s + (0.431 − 0.383i)12-s + (0.225 + 0.274i)13-s + (0.589 − 0.612i)14-s + (−0.304 + 0.126i)15-s + (0.759 + 0.650i)16-s + (1.50 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.999 + 0.0118i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38112 - 0.00819568i\)
\(L(\frac12)\) \(\approx\) \(1.38112 - 0.00819568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.249 + 1.39i)T \)
3 \( 1 + (0.471 - 0.881i)T \)
good5 \( 1 + (-0.986 - 0.809i)T + (0.975 + 4.90i)T^{2} \)
7 \( 1 + (-1.86 - 1.24i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (0.801 - 2.64i)T + (-9.14 - 6.11i)T^{2} \)
13 \( 1 + (-0.812 - 0.989i)T + (-2.53 + 12.7i)T^{2} \)
17 \( 1 + (-6.22 - 2.57i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (2.45 + 0.241i)T + (18.6 + 3.70i)T^{2} \)
23 \( 1 + (-4.61 - 0.917i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-3.05 + 0.926i)T + (24.1 - 16.1i)T^{2} \)
31 \( 1 + (1.35 - 1.35i)T - 31iT^{2} \)
37 \( 1 + (0.392 + 3.98i)T + (-36.2 + 7.21i)T^{2} \)
41 \( 1 + (-0.0227 + 0.114i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-0.604 - 1.13i)T + (-23.8 + 35.7i)T^{2} \)
47 \( 1 + (3.37 - 8.15i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (7.45 + 2.26i)T + (44.0 + 29.4i)T^{2} \)
59 \( 1 + (6.40 - 7.79i)T + (-11.5 - 57.8i)T^{2} \)
61 \( 1 + (0.976 + 0.521i)T + (33.8 + 50.7i)T^{2} \)
67 \( 1 + (-0.00283 - 0.00151i)T + (37.2 + 55.7i)T^{2} \)
71 \( 1 + (0.720 - 1.07i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-5.09 + 3.40i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (5.69 + 13.7i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (0.758 - 7.70i)T + (-81.4 - 16.1i)T^{2} \)
89 \( 1 + (-16.8 + 3.34i)T + (82.2 - 34.0i)T^{2} \)
97 \( 1 + (1.56 - 1.56i)T - 97iT^{2} \)
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

−11.22525958854334950248309449805, −10.48918963329775355453965845708, −9.811668291710731086178619716493, −8.891103030362553449137163141208, −7.85123023315079950567662627836, −6.21170659868905327339899903850, −5.24611764557826380464906700119, −4.35935608130073378480481225502, −3.01487011461153053335841023467, −1.69577825148778245484343440828, 1.05203799271135417318229091075, 3.31500627193320149016148681717, 4.88670931626040109439361260757, 5.53462659350277732287892698339, 6.55783197385309282218352649398, 7.63139500694559145120789881641, 8.250483233024770200708760520002, 9.258303787839630950295269640907, 10.38496411065280567273193318975, 11.42442067713761291949749140850

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