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Properties

Label 2-384-128.29-c1-0-0
Degree $2$
Conductor $384$
Sign $-0.915 - 0.403i$
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  + (−1.28 + 0.600i)2-s + (0.0980 − 0.995i)3-s + (1.27 − 1.53i)4-s + (−1.81 + 0.969i)5-s + (0.472 + 1.33i)6-s + (−0.759 − 3.81i)7-s + (−0.713 + 2.73i)8-s + (−0.980 − 0.195i)9-s + (1.73 − 2.32i)10-s + (−1.72 + 2.10i)11-s + (−1.40 − 1.42i)12-s + (−2.43 + 4.56i)13-s + (3.26 + 4.43i)14-s + (0.786 + 1.89i)15-s + (−0.729 − 3.93i)16-s + (−0.358 + 0.865i)17-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + (0.0565 − 0.574i)3-s + (0.639 − 0.768i)4-s + (−0.810 + 0.433i)5-s + (0.192 + 0.544i)6-s + (−0.287 − 1.44i)7-s + (−0.252 + 0.967i)8-s + (−0.326 − 0.0650i)9-s + (0.550 − 0.736i)10-s + (−0.520 + 0.633i)11-s + (−0.405 − 0.410i)12-s + (−0.676 + 1.26i)13-s + (0.872 + 1.18i)14-s + (0.203 + 0.490i)15-s + (−0.182 − 0.983i)16-s + (−0.0869 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.915 - 0.403i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.915 - 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0206249 + 0.0979078i\)
\(L(\frac12)\) \(\approx\) \(0.0206249 + 0.0979078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.600i)T \)
3 \( 1 + (-0.0980 + 0.995i)T \)
good5 \( 1 + (1.81 - 0.969i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.759 + 3.81i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (1.72 - 2.10i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (2.43 - 4.56i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (0.358 - 0.865i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.294 - 0.969i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (-2.20 + 1.47i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (3.92 - 3.21i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (6.46 - 6.46i)T - 31iT^{2} \)
37 \( 1 + (-1.35 - 0.412i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (2.71 + 4.05i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.592 + 6.01i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (10.6 + 4.40i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (9.08 + 7.45i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (-6.58 - 12.3i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (5.04 + 0.497i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (-11.5 - 1.14i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (9.68 - 1.92i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.33 + 6.70i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-13.5 + 5.61i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.00 + 0.913i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-2.03 - 1.36i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (7.34 - 7.34i)T - 97iT^{2} \)
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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

−11.47037796820361590601006089641, −10.73327322347394925520258533609, −9.931881559395207934154574992576, −8.881871230351847139576908506855, −7.69764556195685425387110095520, −7.14755846770108290687897051791, −6.71749601765619593192299626975, −4.99834767312070238211364644860, −3.57174499905738361638067690385, −1.81243920530379772054409253098, 0.083408047472625042552759247943, 2.54847386231217657509686260555, 3.44915244763223891242629176591, 5.02816526094894155556521384555, 6.08764845688284749536672388726, 7.78333385825024344440552987274, 8.192945846942297956997515578128, 9.254076495698980177177783053870, 9.782048552067838651291123211232, 11.07733593026011132856623331412

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