Properties

Label 2-384-8.3-c6-0-44
Degree $2$
Conductor $384$
Sign $-1$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.5·3-s − 195. i·5-s − 277. i·7-s + 243·9-s − 1.75e3·11-s − 1.24e3i·13-s − 3.04e3i·15-s + 6.88e3·17-s + 4.40e3·19-s − 4.32e3i·21-s − 1.27e4i·23-s − 2.24e4·25-s + 3.78e3·27-s − 8.27e3i·29-s − 4.39e4i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.56i·5-s − 0.809i·7-s + 0.333·9-s − 1.31·11-s − 0.567i·13-s − 0.901i·15-s + 1.40·17-s + 0.641·19-s − 0.467i·21-s − 1.04i·23-s − 1.43·25-s + 0.192·27-s − 0.339i·29-s − 1.47i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-1$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ -1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.878414408\)
\(L(\frac12)\) \(\approx\) \(1.878414408\)
\(L(4)\) 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 \( 1 - 15.5T \)
good5 \( 1 + 195. iT - 1.56e4T^{2} \)
7 \( 1 + 277. iT - 1.17e5T^{2} \)
11 \( 1 + 1.75e3T + 1.77e6T^{2} \)
13 \( 1 + 1.24e3iT - 4.82e6T^{2} \)
17 \( 1 - 6.88e3T + 2.41e7T^{2} \)
19 \( 1 - 4.40e3T + 4.70e7T^{2} \)
23 \( 1 + 1.27e4iT - 1.48e8T^{2} \)
29 \( 1 + 8.27e3iT - 5.94e8T^{2} \)
31 \( 1 + 4.39e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.21e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.47e4T + 4.75e9T^{2} \)
43 \( 1 - 4.54e4T + 6.32e9T^{2} \)
47 \( 1 + 1.52e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.72e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.13e5T + 4.21e10T^{2} \)
61 \( 1 + 8.39e4iT - 5.15e10T^{2} \)
67 \( 1 + 3.73e5T + 9.04e10T^{2} \)
71 \( 1 - 6.67e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.99e5T + 1.51e11T^{2} \)
79 \( 1 - 4.35e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.46e5T + 3.26e11T^{2} \)
89 \( 1 - 8.09e4T + 4.96e11T^{2} \)
97 \( 1 - 8.77e5T + 8.32e11T^{2} \)
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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

−9.906192074040953045044080522552, −8.853670400305541944518620315208, −7.954032299567623887187175938863, −7.53171579892598423665981101924, −5.75653874990142473597925993441, −4.94318451294663136226892825170, −3.96038842548633975967313494233, −2.68338987265155135965706601989, −1.19550638133090670537848892657, −0.39642024436225865404325638332, 1.73314720480577978607540330196, 2.91985305847968993221634815183, 3.32261477761151924592233971145, 5.10713096787248333238137887286, 6.07614660951970506332105382332, 7.28209675111874094546873344442, 7.78413904783142336890477910348, 9.008935957390723786448574366867, 9.993593004250034060939826572780, 10.62199978064500283318289320716

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