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Properties

Label 2-384-16.13-c3-0-19
Degree $2$
Conductor $384$
Sign $0.0362 + 0.999i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 − 2.12i)3-s + (11.8 − 11.8i)5-s + 0.485i·7-s + 8.99i·9-s + (30.9 − 30.9i)11-s + (18.4 + 18.4i)13-s − 50.4·15-s + 135.·17-s + (−65.8 − 65.8i)19-s + (1.02 − 1.02i)21-s + 128. i·23-s − 158. i·25-s + (19.0 − 19.0i)27-s + (−6.64 − 6.64i)29-s − 15.1·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (1.06 − 1.06i)5-s + 0.0261i·7-s + 0.333i·9-s + (0.848 − 0.848i)11-s + (0.393 + 0.393i)13-s − 0.868·15-s + 1.93·17-s + (−0.795 − 0.795i)19-s + (0.0106 − 0.0106i)21-s + 1.16i·23-s − 1.26i·25-s + (0.136 − 0.136i)27-s + (−0.0425 − 0.0425i)29-s − 0.0876·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0362 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.0362 + 0.999i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.0362 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.136795663\)
\(L(\frac12)\) \(\approx\) \(2.136795663\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (-11.8 + 11.8i)T - 125iT^{2} \)
7 \( 1 - 0.485iT - 343T^{2} \)
11 \( 1 + (-30.9 + 30.9i)T - 1.33e3iT^{2} \)
13 \( 1 + (-18.4 - 18.4i)T + 2.19e3iT^{2} \)
17 \( 1 - 135.T + 4.91e3T^{2} \)
19 \( 1 + (65.8 + 65.8i)T + 6.85e3iT^{2} \)
23 \( 1 - 128. iT - 1.21e4T^{2} \)
29 \( 1 + (6.64 + 6.64i)T + 2.43e4iT^{2} \)
31 \( 1 + 15.1T + 2.97e4T^{2} \)
37 \( 1 + (51.4 - 51.4i)T - 5.06e4iT^{2} \)
41 \( 1 + 410. iT - 6.89e4T^{2} \)
43 \( 1 + (69.9 - 69.9i)T - 7.95e4iT^{2} \)
47 \( 1 + 487.T + 1.03e5T^{2} \)
53 \( 1 + (-217. + 217. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-293. + 293. i)T - 2.05e5iT^{2} \)
61 \( 1 + (207. + 207. i)T + 2.26e5iT^{2} \)
67 \( 1 + (284. + 284. i)T + 3.00e5iT^{2} \)
71 \( 1 + 614. iT - 3.57e5T^{2} \)
73 \( 1 + 486. iT - 3.89e5T^{2} \)
79 \( 1 - 960.T + 4.93e5T^{2} \)
83 \( 1 + (463. + 463. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.27e3iT - 7.04e5T^{2} \)
97 \( 1 + 994.T + 9.12e5T^{2} \)
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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

−10.72443584039594118312885903655, −9.612115653109042274818891652749, −8.966750826210112335065993429925, −8.005900879952880043446969076833, −6.66344538427095839781816011721, −5.78212155127286191958281566577, −5.10973887836054338797861456928, −3.59591761865545594182926867143, −1.77727596164219226144824194528, −0.858102994889342053490949599535, 1.46565571144524024345526668830, 2.90925809548828978663644235376, 4.12299038233091916904984561769, 5.55211213742607330375101506089, 6.26337613948397412775262923459, 7.13011946533843442936334152405, 8.432151592437214428146735113052, 9.826692499827417133219627721154, 10.05640855214862338270090835136, 10.90484420684298745009477704146

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