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Properties

Label 2-384-24.11-c3-0-4
Degree $2$
Conductor $384$
Sign $-0.962 + 0.272i$
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  + (1.41 + 5i)3-s + (−23 + 14.1i)9-s − 18i·11-s + 107. i·17-s − 127.·19-s − 125·25-s + (−103. − 95i)27-s + (90 − 25.4i)33-s − 56.5i·41-s − 483.·43-s + 343·49-s + (−537. + 152i)51-s + (−180. − 636. i)57-s − 846i·59-s − 1.09e3·67-s + ⋯
L(s)  = 1  + (0.272 + 0.962i)3-s + (−0.851 + 0.523i)9-s − 0.493i·11-s + 1.53i·17-s − 1.53·19-s − 25-s + (−0.735 − 0.677i)27-s + (0.474 − 0.134i)33-s − 0.215i·41-s − 1.71·43-s + 49-s + (−1.47 + 0.417i)51-s + (−0.418 − 1.47i)57-s − 1.86i·59-s − 1.99·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.962 + 0.272i$
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: \((2,\ 384,\ (\ :3/2),\ -0.962 + 0.272i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6039216764\)
\(L(\frac12)\) \(\approx\) \(0.6039216764\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.41 - 5i)T \)
good5 \( 1 + 125T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 + 18iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 107. iT - 4.91e3T^{2} \)
19 \( 1 + 127.T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 56.5iT - 6.89e4T^{2} \)
43 \( 1 + 483.T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 846iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 1.09e3T + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 430T + 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 - 1.35e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.32e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.91e3T + 9.12e5T^{2} \)
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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

−11.10135323822022868832633039488, −10.52587847387508299353209072959, −9.663407379933159488046959546820, −8.576831839449593899502552472620, −8.097872299186532812899321708964, −6.48448745771862606971275288422, −5.58079139905286779157066356359, −4.32560239626575882758833640778, −3.51666678436270297953652596006, −2.05384414727084546337938854870, 0.18303542954187130978219015007, 1.79593205692884855498555751140, 2.90019593020829020969599279552, 4.39714878560795207867375465301, 5.72773760055005578529838852379, 6.77005270257119978712905172653, 7.50513498166158157610022203528, 8.495147995320005188503102219910, 9.347759857261822726901596808488, 10.42546519361389715902033249415

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