Properties

Label 2-384-4.3-c2-0-10
Degree $2$
Conductor $384$
Sign $i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 4.29·5-s + 2.75i·7-s − 2.99·9-s − 13.7i·11-s − 14.5·13-s − 7.43i·15-s + 22.8·17-s − 16.0i·19-s − 4.77·21-s − 17.1i·23-s − 6.57·25-s − 5.19i·27-s + 21.8·29-s − 38.6i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.858·5-s + 0.393i·7-s − 0.333·9-s − 1.25i·11-s − 1.11·13-s − 0.495i·15-s + 1.34·17-s − 0.844i·19-s − 0.227·21-s − 0.743i·23-s − 0.262·25-s − 0.192i·27-s + 0.754·29-s − 1.24i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.529413 - 0.529413i\)
\(L(\frac12)\) \(\approx\) \(0.529413 - 0.529413i\)
\(L(2)\) 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 \( 1 - 1.73iT \)
good5 \( 1 + 4.29T + 25T^{2} \)
7 \( 1 - 2.75iT - 49T^{2} \)
11 \( 1 + 13.7iT - 121T^{2} \)
13 \( 1 + 14.5T + 169T^{2} \)
17 \( 1 - 22.8T + 289T^{2} \)
19 \( 1 + 16.0iT - 361T^{2} \)
23 \( 1 + 17.1iT - 529T^{2} \)
29 \( 1 - 21.8T + 841T^{2} \)
31 \( 1 + 38.6iT - 961T^{2} \)
37 \( 1 + 66.4T + 1.36e3T^{2} \)
41 \( 1 - 23.2T + 1.68e3T^{2} \)
43 \( 1 + 47.9iT - 1.84e3T^{2} \)
47 \( 1 + 14.8iT - 2.20e3T^{2} \)
53 \( 1 + 65.5T + 2.80e3T^{2} \)
59 \( 1 - 65.8iT - 3.48e3T^{2} \)
61 \( 1 - 40.1T + 3.72e3T^{2} \)
67 \( 1 + 74.8iT - 4.48e3T^{2} \)
71 \( 1 + 122. iT - 5.04e3T^{2} \)
73 \( 1 + 144.T + 5.32e3T^{2} \)
79 \( 1 - 128. iT - 6.24e3T^{2} \)
83 \( 1 + 22.0iT - 6.88e3T^{2} \)
89 \( 1 + 122.T + 7.92e3T^{2} \)
97 \( 1 + 88.3T + 9.40e3T^{2} \)
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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

−10.90408708989971924444634879701, −10.04466751312845874985533196009, −9.001356840326585867768580553336, −8.196752767566621270160141727265, −7.27291589523722248802686504807, −5.90524763752171809285178136454, −4.95346177231044835112952091348, −3.77239627911664169032447860409, −2.73781710422507282641429163482, −0.33762254544829742289622063099, 1.52797980866440134138739861454, 3.15795273391272359209945141411, 4.39224442054104935801887489634, 5.49847563534725568695769592865, 7.01022146554197453419130685034, 7.48683988543571954501379001037, 8.285630949424002549973506114337, 9.704777955352381772683983520406, 10.31881818890008902789779242623, 11.65139785645475308532014413332

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