Properties

Label 2-384-384.107-c1-0-9
Degree $2$
Conductor $384$
Sign $-0.981 + 0.189i$
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  + (−0.982 + 1.01i)2-s + (0.993 + 1.41i)3-s + (−0.0684 − 1.99i)4-s + (−2.02 + 0.613i)5-s + (−2.41 − 0.383i)6-s + (0.135 − 0.682i)7-s + (2.10 + 1.89i)8-s + (−1.02 + 2.81i)9-s + (1.36 − 2.65i)10-s + (−0.318 + 3.22i)11-s + (2.76 − 2.08i)12-s + (−2.17 − 0.659i)13-s + (0.560 + 0.808i)14-s + (−2.87 − 2.25i)15-s + (−3.99 + 0.273i)16-s + (−3.85 + 1.59i)17-s + ⋯
L(s)  = 1  + (−0.694 + 0.719i)2-s + (0.573 + 0.819i)3-s + (−0.0342 − 0.999i)4-s + (−0.903 + 0.274i)5-s + (−0.987 − 0.156i)6-s + (0.0513 − 0.257i)7-s + (0.742 + 0.669i)8-s + (−0.341 + 0.939i)9-s + (0.430 − 0.840i)10-s + (−0.0959 + 0.973i)11-s + (0.798 − 0.601i)12-s + (−0.603 − 0.182i)13-s + (0.149 + 0.216i)14-s + (−0.743 − 0.582i)15-s + (−0.997 + 0.0684i)16-s + (−0.935 + 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.981 + 0.189i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0567511 - 0.592488i\)
\(L(\frac12)\) \(\approx\) \(0.0567511 - 0.592488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.982 - 1.01i)T \)
3 \( 1 + (-0.993 - 1.41i)T \)
good5 \( 1 + (2.02 - 0.613i)T + (4.15 - 2.77i)T^{2} \)
7 \( 1 + (-0.135 + 0.682i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.318 - 3.22i)T + (-10.7 - 2.14i)T^{2} \)
13 \( 1 + (2.17 + 0.659i)T + (10.8 + 7.22i)T^{2} \)
17 \( 1 + (3.85 - 1.59i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (4.05 - 2.16i)T + (10.5 - 15.7i)T^{2} \)
23 \( 1 + (-1.31 + 1.96i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.0407 - 0.413i)T + (-28.4 + 5.65i)T^{2} \)
31 \( 1 + (-3.43 + 3.43i)T - 31iT^{2} \)
37 \( 1 + (-8.03 - 4.29i)T + (20.5 + 30.7i)T^{2} \)
41 \( 1 + (-1.97 + 2.95i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (2.30 + 2.81i)T + (-8.38 + 42.1i)T^{2} \)
47 \( 1 + (7.55 - 3.12i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (0.935 - 9.50i)T + (-51.9 - 10.3i)T^{2} \)
59 \( 1 + (9.03 - 2.73i)T + (49.0 - 32.7i)T^{2} \)
61 \( 1 + (-0.704 + 0.858i)T + (-11.9 - 59.8i)T^{2} \)
67 \( 1 + (-9.31 - 7.64i)T + (13.0 + 65.7i)T^{2} \)
71 \( 1 + (-2.71 + 13.6i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-7.31 + 1.45i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (5.05 - 12.2i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.15 - 2.21i)T + (46.1 - 69.0i)T^{2} \)
89 \( 1 + (-0.450 + 0.301i)T + (34.0 - 82.2i)T^{2} \)
97 \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \)
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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

−11.41215144836579507822459186701, −10.61816428232860608708437460308, −9.919379217399980508159381051268, −9.010737454716854634886369851343, −8.036792301537716271546554104668, −7.49160297053032199186321093585, −6.33203050055387886736491927756, −4.79546814300421765136866120045, −4.12500342273551330825872719730, −2.34297414663750688352530997745, 0.44449364002226324199809138521, 2.22577735142154571034750313938, 3.31768358807874562293565577042, 4.53617440115239368114292754825, 6.45036540557223237967594401884, 7.42637280740174252791915738678, 8.307390029271174845089441614593, 8.779481267639775774015532683765, 9.768435893049727692382638697765, 11.26025109173988319584069903760

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