Properties

Label 2-384-128.125-c1-0-27
Degree $2$
Conductor $384$
Sign $0.933 + 0.357i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 − 0.748i)2-s + (0.634 + 0.773i)3-s + (0.878 − 1.79i)4-s + (3.32 + 1.00i)5-s + (1.33 + 0.452i)6-s + (−0.658 + 0.131i)7-s + (−0.290 − 2.81i)8-s + (−0.195 + 0.980i)9-s + (4.74 − 1.27i)10-s + (−4.01 + 0.395i)11-s + (1.94 − 0.460i)12-s + (0.0924 + 0.304i)13-s + (−0.692 + 0.650i)14-s + (1.32 + 3.20i)15-s + (−2.45 − 3.15i)16-s + (−0.921 + 2.22i)17-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.366 + 0.446i)3-s + (0.439 − 0.898i)4-s + (1.48 + 0.451i)5-s + (0.546 + 0.184i)6-s + (−0.249 + 0.0495i)7-s + (−0.102 − 0.994i)8-s + (−0.0650 + 0.326i)9-s + (1.50 − 0.404i)10-s + (−1.21 + 0.119i)11-s + (0.561 − 0.132i)12-s + (0.0256 + 0.0845i)13-s + (−0.185 + 0.173i)14-s + (0.343 + 0.828i)15-s + (−0.613 − 0.789i)16-s + (−0.223 + 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.933 + 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67444 - 0.494176i\)
\(L(\frac12)\) \(\approx\) \(2.67444 - 0.494176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.748i)T \)
3 \( 1 + (-0.634 - 0.773i)T \)
good5 \( 1 + (-3.32 - 1.00i)T + (4.15 + 2.77i)T^{2} \)
7 \( 1 + (0.658 - 0.131i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (4.01 - 0.395i)T + (10.7 - 2.14i)T^{2} \)
13 \( 1 + (-0.0924 - 0.304i)T + (-10.8 + 7.22i)T^{2} \)
17 \( 1 + (0.921 - 2.22i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (1.18 + 0.635i)T + (10.5 + 15.7i)T^{2} \)
23 \( 1 + (2.01 + 3.01i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.170 - 1.73i)T + (-28.4 - 5.65i)T^{2} \)
31 \( 1 + (-0.247 + 0.247i)T - 31iT^{2} \)
37 \( 1 + (4.20 + 7.86i)T + (-20.5 + 30.7i)T^{2} \)
41 \( 1 + (-9.34 + 6.24i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-0.863 + 1.05i)T + (-8.38 - 42.1i)T^{2} \)
47 \( 1 + (-6.15 - 2.55i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-1.11 - 11.2i)T + (-51.9 + 10.3i)T^{2} \)
59 \( 1 + (-3.13 + 10.3i)T + (-49.0 - 32.7i)T^{2} \)
61 \( 1 + (10.9 - 8.96i)T + (11.9 - 59.8i)T^{2} \)
67 \( 1 + (-0.610 + 0.501i)T + (13.0 - 65.7i)T^{2} \)
71 \( 1 + (-1.19 - 6.01i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-5.01 - 0.996i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (15.3 - 6.34i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (5.70 - 10.6i)T + (-46.1 - 69.0i)T^{2} \)
89 \( 1 + (-3.08 + 4.61i)T + (-34.0 - 82.2i)T^{2} \)
97 \( 1 + (-6.58 + 6.58i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90688035455452576557078312115, −10.52676199505508765045467219089, −9.770008884560161736255559223325, −8.890968727645560175426944176764, −7.29577787080547539520459839623, −6.09194915467799140067723460454, −5.48098839499863524173247441428, −4.25999320570604050978825695054, −2.82718322134074021680236870849, −2.09200142571062442276789897976, 2.03529685905101548200220927885, 3.06736233481105091581129925434, 4.75846002024906683077478014449, 5.66934334655911488589693381432, 6.40003330848759489679704227267, 7.53727772416472476945562335853, 8.446514067915358796571237229344, 9.470501557835473664562315967578, 10.41893526606137260338591891983, 11.67978289174938008084577798498

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