Properties

Label 2-384-24.11-c3-0-39
Degree $2$
Conductor $384$
Sign $-0.290 + 0.956i$
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  + (2.44 + 4.58i)3-s − 2.31·5-s − 29.6i·7-s + (−15 + 22.4i)9-s + 22.8i·11-s − 8.66i·13-s + (−5.66 − 10.6i)15-s − 93.3i·17-s − 85.4·19-s + (135. − 72.6i)21-s − 116.·23-s − 119.·25-s + (−139. − 13.7i)27-s − 103.·29-s + 10.6i·31-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)3-s − 0.207·5-s − 1.60i·7-s + (−0.555 + 0.831i)9-s + 0.625i·11-s − 0.184i·13-s + (−0.0975 − 0.182i)15-s − 1.33i·17-s − 1.03·19-s + (1.41 − 0.755i)21-s − 1.05·23-s − 0.957·25-s + (−0.995 − 0.0979i)27-s − 0.662·29-s + 0.0614i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.290 + 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8853803139\)
\(L(\frac12)\) \(\approx\) \(0.8853803139\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.44 - 4.58i)T \)
good5 \( 1 + 2.31T + 125T^{2} \)
7 \( 1 + 29.6iT - 343T^{2} \)
11 \( 1 - 22.8iT - 1.33e3T^{2} \)
13 \( 1 + 8.66iT - 2.19e3T^{2} \)
17 \( 1 + 93.3iT - 4.91e3T^{2} \)
19 \( 1 + 85.4T + 6.85e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 - 10.6iT - 2.97e4T^{2} \)
37 \( 1 + 380. iT - 5.06e4T^{2} \)
41 \( 1 + 257. iT - 6.89e4T^{2} \)
43 \( 1 - 359.T + 7.95e4T^{2} \)
47 \( 1 - 293.T + 1.03e5T^{2} \)
53 \( 1 - 679.T + 1.48e5T^{2} \)
59 \( 1 - 45.8iT - 2.05e5T^{2} \)
61 \( 1 + 595. iT - 2.26e5T^{2} \)
67 \( 1 + 206.T + 3.00e5T^{2} \)
71 \( 1 + 996.T + 3.57e5T^{2} \)
73 \( 1 + 593.T + 3.89e5T^{2} \)
79 \( 1 - 910. iT - 4.93e5T^{2} \)
83 \( 1 - 332. iT - 5.71e5T^{2} \)
89 \( 1 - 821. iT - 7.04e5T^{2} \)
97 \( 1 - 420.T + 9.12e5T^{2} \)
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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

−10.51768646166640746086758425239, −9.856360266927136774095488504033, −8.954065228250439074366999597428, −7.70365701269539751484029201156, −7.20416299965730794464161393534, −5.60541030640563978862970585865, −4.26751901753234692874637441823, −3.89308984050703027655006004632, −2.28706216497373339454775517235, −0.26453521424781961298245148793, 1.72631015693474768333407764614, 2.69859829696648727389772793429, 4.02488526074733755609183165296, 5.88955573795951463806507885789, 6.14133389421716299519781644811, 7.64408192591979163812349114312, 8.519301196027412857556618816213, 8.924022183824225753139877470122, 10.22806910786470177981546215593, 11.57286346153625284928903426549

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