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Properties

Label 2-384-48.35-c3-0-27
Degree $2$
Conductor $384$
Sign $0.572 - 0.820i$
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  + (4.66 + 2.28i)3-s + (11.5 + 11.5i)5-s + 0.829·7-s + (16.5 + 21.3i)9-s + (38.2 − 38.2i)11-s + (20.0 + 20.0i)13-s + (27.4 + 80.0i)15-s − 77.9i·17-s + (37.6 − 37.6i)19-s + (3.87 + 1.89i)21-s + 159. i·23-s + 140. i·25-s + (28.6 + 137. i)27-s + (−26.6 + 26.6i)29-s − 226. i·31-s + ⋯
L(s)  = 1  + (0.898 + 0.439i)3-s + (1.02 + 1.02i)5-s + 0.0447·7-s + (0.613 + 0.789i)9-s + (1.04 − 1.04i)11-s + (0.427 + 0.427i)13-s + (0.472 + 1.37i)15-s − 1.11i·17-s + (0.455 − 0.455i)19-s + (0.0402 + 0.0196i)21-s + 1.44i·23-s + 1.12i·25-s + (0.204 + 0.978i)27-s + (−0.170 + 0.170i)29-s − 1.31i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.572 - 0.820i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.572 - 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.441942101\)
\(L(\frac12)\) \(\approx\) \(3.441942101\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.66 - 2.28i)T \)
good5 \( 1 + (-11.5 - 11.5i)T + 125iT^{2} \)
7 \( 1 - 0.829T + 343T^{2} \)
11 \( 1 + (-38.2 + 38.2i)T - 1.33e3iT^{2} \)
13 \( 1 + (-20.0 - 20.0i)T + 2.19e3iT^{2} \)
17 \( 1 + 77.9iT - 4.91e3T^{2} \)
19 \( 1 + (-37.6 + 37.6i)T - 6.85e3iT^{2} \)
23 \( 1 - 159. iT - 1.21e4T^{2} \)
29 \( 1 + (26.6 - 26.6i)T - 2.43e4iT^{2} \)
31 \( 1 + 226. iT - 2.97e4T^{2} \)
37 \( 1 + (176. - 176. i)T - 5.06e4iT^{2} \)
41 \( 1 + 301.T + 6.89e4T^{2} \)
43 \( 1 + (-96.1 - 96.1i)T + 7.95e4iT^{2} \)
47 \( 1 + 122.T + 1.03e5T^{2} \)
53 \( 1 + (138. + 138. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-168. + 168. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-178. - 178. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-233. + 233. i)T - 3.00e5iT^{2} \)
71 \( 1 + 668. iT - 3.57e5T^{2} \)
73 \( 1 - 1.21e3iT - 3.89e5T^{2} \)
79 \( 1 - 39.4iT - 4.93e5T^{2} \)
83 \( 1 + (-293. - 293. i)T + 5.71e5iT^{2} \)
89 \( 1 + 732.T + 7.04e5T^{2} \)
97 \( 1 - 451.T + 9.12e5T^{2} \)
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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

−11.09144013624748434944122658576, −9.783714345074460238995708794242, −9.485682789061896499074436854346, −8.464076655844917867667669666941, −7.22466143705937158591557054620, −6.38811396780407332717000454449, −5.23776576104535444095897532245, −3.69217913528413819404716666760, −2.89695995865521484850422338918, −1.60267390998114960894463568328, 1.26146160309985536090265413853, 1.98924509596010957574038386821, 3.64450287776278107316705151545, 4.79900059115077161303777055050, 6.08245528859756953631635750649, 6.96370039163724353581498712976, 8.289246341227089428194774522214, 8.850986986028995655698244561573, 9.686218207269449967747432614445, 10.44825347604710902379807538718

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