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Properties

Label 2-384-48.35-c3-0-36
Degree $2$
Conductor $384$
Sign $-0.341 + 0.939i$
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  + (1.96 − 4.81i)3-s + (−6.30 − 6.30i)5-s + 24.6·7-s + (−19.3 − 18.8i)9-s + (40.4 − 40.4i)11-s + (47.3 + 47.3i)13-s + (−42.6 + 17.9i)15-s − 41.7i·17-s + (−10.6 + 10.6i)19-s + (48.3 − 118. i)21-s + 53.4i·23-s − 45.5i·25-s + (−128. + 55.9i)27-s + (105. − 105. i)29-s + 3.14i·31-s + ⋯
L(s)  = 1  + (0.377 − 0.926i)3-s + (−0.563 − 0.563i)5-s + 1.33·7-s + (−0.715 − 0.698i)9-s + (1.10 − 1.10i)11-s + (1.00 + 1.00i)13-s + (−0.734 + 0.309i)15-s − 0.595i·17-s + (−0.128 + 0.128i)19-s + (0.502 − 1.23i)21-s + 0.484i·23-s − 0.364i·25-s + (−0.917 + 0.398i)27-s + (0.674 − 0.674i)29-s + 0.0182i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.341 + 0.939i$
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: \((2,\ 384,\ (\ :3/2),\ -0.341 + 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.337265302\)
\(L(\frac12)\) \(\approx\) \(2.337265302\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.96 + 4.81i)T \)
good5 \( 1 + (6.30 + 6.30i)T + 125iT^{2} \)
7 \( 1 - 24.6T + 343T^{2} \)
11 \( 1 + (-40.4 + 40.4i)T - 1.33e3iT^{2} \)
13 \( 1 + (-47.3 - 47.3i)T + 2.19e3iT^{2} \)
17 \( 1 + 41.7iT - 4.91e3T^{2} \)
19 \( 1 + (10.6 - 10.6i)T - 6.85e3iT^{2} \)
23 \( 1 - 53.4iT - 1.21e4T^{2} \)
29 \( 1 + (-105. + 105. i)T - 2.43e4iT^{2} \)
31 \( 1 - 3.14iT - 2.97e4T^{2} \)
37 \( 1 + (42.1 - 42.1i)T - 5.06e4iT^{2} \)
41 \( 1 - 152.T + 6.89e4T^{2} \)
43 \( 1 + (221. + 221. i)T + 7.95e4iT^{2} \)
47 \( 1 + 381.T + 1.03e5T^{2} \)
53 \( 1 + (294. + 294. i)T + 1.48e5iT^{2} \)
59 \( 1 + (445. - 445. i)T - 2.05e5iT^{2} \)
61 \( 1 + (21.8 + 21.8i)T + 2.26e5iT^{2} \)
67 \( 1 + (-572. + 572. i)T - 3.00e5iT^{2} \)
71 \( 1 - 612. iT - 3.57e5T^{2} \)
73 \( 1 - 331. iT - 3.89e5T^{2} \)
79 \( 1 + 427. iT - 4.93e5T^{2} \)
83 \( 1 + (-245. - 245. i)T + 5.71e5iT^{2} \)
89 \( 1 + 188.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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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

−11.15534595380159661011884369798, −9.331127423847103019455463913799, −8.449495398874918084853779706939, −8.166746691272321850012543680695, −6.89294376479595149842768204017, −5.97734851113181608436141829078, −4.57642629187475559476256341729, −3.51572105944084610269373455548, −1.76483003705546352967757722483, −0.847514773192127255546944145091, 1.61458538009387961812033604819, 3.23020015598247748409596589133, 4.20821621810785237032990185409, 5.04937140223686474700390082266, 6.44852602976066082911309593699, 7.75384523762054189797638639021, 8.363763623011891048844101876124, 9.342107561211348665500170187906, 10.49054760695504557832103508738, 11.01963276598452362907609216022

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