↑

Properties

Label 2-384-128.101-c1-0-23
Degree $2$
Conductor $384$
Sign $0.472 + 0.881i$
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  + (1.26 − 0.640i)2-s + (−0.881 + 0.471i)3-s + (1.17 − 1.61i)4-s + (−0.276 − 0.336i)5-s + (−0.810 + 1.15i)6-s + (0.338 − 0.226i)7-s + (0.453 − 2.79i)8-s + (0.555 − 0.831i)9-s + (−0.563 − 0.247i)10-s + (5.34 − 1.62i)11-s + (−0.279 + 1.98i)12-s + (−0.687 − 0.563i)13-s + (0.282 − 0.502i)14-s + (0.402 + 0.166i)15-s + (−1.21 − 3.81i)16-s + (−2.42 + 1.00i)17-s + ⋯
L(s)  = 1  + (0.891 − 0.452i)2-s + (−0.509 + 0.272i)3-s + (0.589 − 0.807i)4-s + (−0.123 − 0.150i)5-s + (−0.330 + 0.473i)6-s + (0.128 − 0.0855i)7-s + (0.160 − 0.987i)8-s + (0.185 − 0.277i)9-s + (−0.178 − 0.0782i)10-s + (1.61 − 0.488i)11-s + (−0.0805 + 0.571i)12-s + (−0.190 − 0.156i)13-s + (0.0754 − 0.134i)14-s + (0.103 + 0.0430i)15-s + (−0.304 − 0.952i)16-s + (−0.588 + 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.472 + 0.881i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71768 - 1.02861i\)
\(L(\frac12)\) \(\approx\) \(1.71768 - 1.02861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.640i)T \)
3 \( 1 + (0.881 - 0.471i)T \)
good5 \( 1 + (0.276 + 0.336i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-0.338 + 0.226i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-5.34 + 1.62i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (0.687 + 0.563i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (2.42 - 1.00i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.274 + 2.78i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (-2.38 + 0.473i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (2.19 - 7.25i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-6.54 - 6.54i)T + 31iT^{2} \)
37 \( 1 + (8.61 + 0.848i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-1.68 - 8.49i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.39 + 1.81i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-2.17 - 5.24i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-0.0205 - 0.0676i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (2.24 - 1.83i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (2.98 + 5.57i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (3.71 + 6.94i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-5.59 - 8.37i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-8.63 - 5.77i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (1.25 - 3.04i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.917 - 0.0903i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (6.22 + 1.23i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (9.59 + 9.59i)T + 97iT^{2} \)
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

−11.22390929343920410955335522145, −10.67255027326099615256093728791, −9.531282133572055418663463732269, −8.625034238772635646626421664175, −6.88018937124325631810482526468, −6.36143777730162562192067743707, −5.06696768377045298196722661284, −4.27975799144668538765972430550, −3.13488094841638045860195362219, −1.27350750702614013575895151030, 1.96869811337174984419513361271, 3.69367293231081711887280676495, 4.61704877775419918456229344541, 5.78657049160035168084792881032, 6.69555536542559108118245202835, 7.33552634347908189459188218084, 8.536654191184328817787588521015, 9.660689785883838190516042194626, 11.00543654857144247144752977629, 11.80535506581929034721007432287

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