Properties

Label 2-384-128.93-c1-0-22
Degree $2$
Conductor $384$
Sign $-0.999 + 0.0337i$
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  + (−0.0761 − 1.41i)2-s + (−0.995 − 0.0980i)3-s + (−1.98 + 0.215i)4-s + (1.41 + 2.64i)5-s + (−0.0625 + 1.41i)6-s + (−0.878 − 4.41i)7-s + (0.455 + 2.79i)8-s + (0.980 + 0.195i)9-s + (3.62 − 2.19i)10-s + (−3.15 − 2.58i)11-s + (1.99 − 0.0192i)12-s + (−2.78 − 1.48i)13-s + (−6.16 + 1.57i)14-s + (−1.14 − 2.76i)15-s + (3.90 − 0.855i)16-s + (2.08 − 5.03i)17-s + ⋯
L(s)  = 1  + (−0.0538 − 0.998i)2-s + (−0.574 − 0.0565i)3-s + (−0.994 + 0.107i)4-s + (0.631 + 1.18i)5-s + (−0.0255 + 0.576i)6-s + (−0.332 − 1.66i)7-s + (0.161 + 0.986i)8-s + (0.326 + 0.0650i)9-s + (1.14 − 0.694i)10-s + (−0.951 − 0.780i)11-s + (0.577 − 0.00556i)12-s + (−0.772 − 0.412i)13-s + (−1.64 + 0.421i)14-s + (−0.296 − 0.715i)15-s + (0.976 − 0.213i)16-s + (0.506 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.999 + 0.0337i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.999 + 0.0337i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00995644 - 0.590133i\)
\(L(\frac12)\) \(\approx\) \(0.00995644 - 0.590133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0761 + 1.41i)T \)
3 \( 1 + (0.995 + 0.0980i)T \)
good5 \( 1 + (-1.41 - 2.64i)T + (-2.77 + 4.15i)T^{2} \)
7 \( 1 + (0.878 + 4.41i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (3.15 + 2.58i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (2.78 + 1.48i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (-2.08 + 5.03i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (7.63 - 2.31i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (-1.59 + 1.06i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (3.94 + 4.81i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (0.500 - 0.500i)T - 31iT^{2} \)
37 \( 1 + (1.25 - 4.13i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (-1.39 - 2.08i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-5.81 + 0.572i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (1.03 + 0.427i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.146 - 0.178i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (-7.57 + 4.04i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (-0.985 + 10.0i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (0.886 - 9.00i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (-3.44 + 0.686i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.10 + 5.53i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-4.65 + 1.92i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (3.00 + 9.90i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (1.97 + 1.31i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-9.92 + 9.92i)T - 97iT^{2} \)
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

−10.67854017081370829272496453028, −10.34094499016057913032467078548, −9.667475439286545932500906277462, −7.995938257081839141872999667415, −7.13066184528700418368893756391, −6.02336028887779380001250116837, −4.78923328551516562410215281570, −3.52470144387367955513381190474, −2.44616669792081842431805765837, −0.41199533893442295183676937219, 2.07749354459203634008457516817, 4.41050444688281041591937813141, 5.37055219344052404691147963367, 5.74795226529845028376940450004, 6.90938238087888435529822467897, 8.230925661501450492803190059009, 9.009819508459638556655964048897, 9.588449362431435826058629775467, 10.65163204045282425551760979640, 12.39880375877674059590486046424

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