Properties

Label 2-384-8.5-c1-0-1
Degree $2$
Conductor $384$
Sign $-i$
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  i·3-s + 2.82i·5-s − 2.82·7-s − 9-s + 4i·11-s + 5.65i·13-s + 2.82·15-s − 2·17-s − 4i·19-s + 2.82i·21-s + 5.65·23-s − 3.00·25-s + i·27-s + 2.82i·29-s − 8.48·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.26i·5-s − 1.06·7-s − 0.333·9-s + 1.20i·11-s + 1.56i·13-s + 0.730·15-s − 0.485·17-s − 0.917i·19-s + 0.617i·21-s + 1.17·23-s − 0.600·25-s + 0.192i·27-s + 0.525i·29-s − 1.52·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681378 + 0.681378i\)
\(L(\frac12)\) \(\approx\) \(0.681378 + 0.681378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 5.65iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 2.82iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−11.39651086857171981447994480627, −10.86214624597115501074020938941, −9.589048740438077970213390282372, −9.091707865710564112291081055953, −7.31227510997071894143389406489, −6.94803412047455459878481713307, −6.26220373076718752600619059946, −4.59014808961757648683749422747, −3.19584874368916845285221376212, −2.12688548092163566730867907666, 0.63161969456271573036383893493, 3.01250857646344909223200469975, 4.01576280278593421328541002766, 5.42757163728111299870223051784, 5.89665677343569916421534970387, 7.51160963545188632611326604934, 8.694247754189887861826088654208, 9.079249102634239362327622980988, 10.21624786401804013769402092726, 10.92306869392882484147080324064

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