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Properties

Label 2-384-48.5-c2-0-27
Degree $2$
Conductor $384$
Sign $-0.860 + 0.509i$
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  + (0.164 − 2.99i)3-s + (3.61 − 3.61i)5-s − 12.2i·7-s + (−8.94 − 0.985i)9-s + (−1.76 + 1.76i)11-s + (2.38 − 2.38i)13-s + (−10.2 − 11.4i)15-s + 20.0i·17-s + (8.77 − 8.77i)19-s + (−36.7 − 2.02i)21-s − 13.1·23-s − 1.10i·25-s + (−4.42 + 26.6i)27-s + (6.51 + 6.51i)29-s + 37.5·31-s + ⋯
L(s)  = 1  + (0.0548 − 0.998i)3-s + (0.722 − 0.722i)5-s − 1.75i·7-s + (−0.993 − 0.109i)9-s + (−0.160 + 0.160i)11-s + (0.183 − 0.183i)13-s + (−0.681 − 0.761i)15-s + 1.18i·17-s + (0.461 − 0.461i)19-s + (−1.75 − 0.0962i)21-s − 0.573·23-s − 0.0443i·25-s + (−0.163 + 0.986i)27-s + (0.224 + 0.224i)29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.442710 - 1.61654i\)
\(L(\frac12)\) \(\approx\) \(0.442710 - 1.61654i\)
\(L(2)\) 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 \( 1 + (-0.164 + 2.99i)T \)
good5 \( 1 + (-3.61 + 3.61i)T - 25iT^{2} \)
7 \( 1 + 12.2iT - 49T^{2} \)
11 \( 1 + (1.76 - 1.76i)T - 121iT^{2} \)
13 \( 1 + (-2.38 + 2.38i)T - 169iT^{2} \)
17 \( 1 - 20.0iT - 289T^{2} \)
19 \( 1 + (-8.77 + 8.77i)T - 361iT^{2} \)
23 \( 1 + 13.1T + 529T^{2} \)
29 \( 1 + (-6.51 - 6.51i)T + 841iT^{2} \)
31 \( 1 - 37.5T + 961T^{2} \)
37 \( 1 + (10.0 + 10.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 4.57T + 1.68e3T^{2} \)
43 \( 1 + (21.2 + 21.2i)T + 1.84e3iT^{2} \)
47 \( 1 + 54.8iT - 2.20e3T^{2} \)
53 \( 1 + (21.5 - 21.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-53.6 + 53.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-19.2 + 19.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (31.5 - 31.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 65.1T + 5.04e3T^{2} \)
73 \( 1 + 50.2iT - 5.32e3T^{2} \)
79 \( 1 - 20.9T + 6.24e3T^{2} \)
83 \( 1 + (-6.35 - 6.35i)T + 6.88e3iT^{2} \)
89 \( 1 - 166.T + 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−10.65140886446382212693697789590, −9.955247111049603510849521919709, −8.721956659196708546784238474006, −7.87349005172194184411444444136, −6.98926103121563945609858288439, −6.08310073206766080808105343495, −4.89013453238472163607897629205, −3.55813478925109419331498255480, −1.79526889661747492982981382900, −0.74243228896546572872097761396, 2.37899908781492606395560620987, 3.10866779040393195016165644962, 4.78218701605451466017351353298, 5.71628874420118574923776442140, 6.38841759297082618526530376916, 8.051227543437436041562553363323, 8.984954414249851622252367027826, 9.692269603250032525260832426951, 10.38881046473152572571499773019, 11.59600133274423464066115188940

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