Properties

Label 2-384-32.21-c1-0-2
Degree $2$
Conductor $384$
Sign $0.487 - 0.873i$
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  + (−0.923 + 0.382i)3-s + (−0.00259 + 0.00626i)5-s + (2.41 + 2.41i)7-s + (0.707 − 0.707i)9-s + (−1.29 − 0.538i)11-s + (−0.559 − 1.35i)13-s − 0.00678i·15-s + 5.82i·17-s + (2.67 + 6.46i)19-s + (−3.16 − 1.30i)21-s + (−0.178 + 0.178i)23-s + (3.53 + 3.53i)25-s + (−0.382 + 0.923i)27-s + (5.72 − 2.37i)29-s + 6.19·31-s + ⋯
L(s)  = 1  + (−0.533 + 0.220i)3-s + (−0.00116 + 0.00280i)5-s + (0.914 + 0.914i)7-s + (0.235 − 0.235i)9-s + (−0.391 − 0.162i)11-s + (−0.155 − 0.374i)13-s − 0.00175i·15-s + 1.41i·17-s + (0.614 + 1.48i)19-s + (−0.689 − 0.285i)21-s + (−0.0372 + 0.0372i)23-s + (0.707 + 0.707i)25-s + (−0.0736 + 0.177i)27-s + (1.06 − 0.440i)29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.487 - 0.873i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.487 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02963 + 0.604310i\)
\(L(\frac12)\) \(\approx\) \(1.02963 + 0.604310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.923 - 0.382i)T \)
good5 \( 1 + (0.00259 - 0.00626i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 + (1.29 + 0.538i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.559 + 1.35i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.82iT - 17T^{2} \)
19 \( 1 + (-2.67 - 6.46i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.178 - 0.178i)T - 23iT^{2} \)
29 \( 1 + (-5.72 + 2.37i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.19T + 31T^{2} \)
37 \( 1 + (2.02 - 4.89i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.36 - 3.36i)T - 41iT^{2} \)
43 \( 1 + (9.37 + 3.88i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 12.5iT - 47T^{2} \)
53 \( 1 + (8.36 + 3.46i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.59 - 3.85i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-7.27 + 3.01i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-4.38 + 1.81i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.95 + 5.95i)T + 71iT^{2} \)
73 \( 1 + (-7.85 + 7.85i)T - 73iT^{2} \)
79 \( 1 - 1.42iT - 79T^{2} \)
83 \( 1 + (-3.03 - 7.32i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.96 + 9.96i)T + 89iT^{2} \)
97 \( 1 + 1.24T + 97T^{2} \)
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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

−11.64783204804786279139243791070, −10.51202581437344530215711615072, −9.940111889588061224674313023864, −8.449857736870647080889229654639, −8.094961380757725143289707730878, −6.55549799010015521885475292628, −5.57209035983364346450872724870, −4.85026561750379174265935856913, −3.37220124460587096571728736941, −1.69324211746900086920120349837, 0.945350280244579054022604331471, 2.74259004967434068027331442189, 4.61649989713715166164926294089, 5.00412768839378400532067278033, 6.65592694518836226811795573758, 7.28304836370425658821786320603, 8.259382034728217635134949658242, 9.462147891608497397731887741325, 10.45644068556488203113610731308, 11.24443052739392037424559694393

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