Properties

Label 2-384-12.11-c5-0-69
Degree $2$
Conductor $384$
Sign $-0.244 + 0.969i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (15.1 + 3.81i)3-s + 40.5i·5-s − 81.4i·7-s + (213. + 115. i)9-s − 499.·11-s − 262.·13-s + (−154. + 613. i)15-s − 565. i·17-s − 2.13e3i·19-s + (310. − 1.23e3i)21-s − 3.65e3·23-s + 1.47e3·25-s + (2.79e3 + 2.55e3i)27-s − 7.31e3i·29-s + 2.21e3i·31-s + ⋯
L(s)  = 1  + (0.969 + 0.244i)3-s + 0.725i·5-s − 0.628i·7-s + (0.880 + 0.474i)9-s − 1.24·11-s − 0.430·13-s + (−0.177 + 0.703i)15-s − 0.474i·17-s − 1.35i·19-s + (0.153 − 0.609i)21-s − 1.43·23-s + 0.473·25-s + (0.737 + 0.675i)27-s − 1.61i·29-s + 0.414i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.244 + 0.969i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.244 + 0.969i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.356154051\)
\(L(\frac12)\) \(\approx\) \(1.356154051\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.1 - 3.81i)T \)
good5 \( 1 - 40.5iT - 3.12e3T^{2} \)
7 \( 1 + 81.4iT - 1.68e4T^{2} \)
11 \( 1 + 499.T + 1.61e5T^{2} \)
13 \( 1 + 262.T + 3.71e5T^{2} \)
17 \( 1 + 565. iT - 1.41e6T^{2} \)
19 \( 1 + 2.13e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.65e3T + 6.43e6T^{2} \)
29 \( 1 + 7.31e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.21e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.00e4T + 6.93e7T^{2} \)
41 \( 1 + 2.86e3iT - 1.15e8T^{2} \)
43 \( 1 + 8.28e3iT - 1.47e8T^{2} \)
47 \( 1 + 3.92e3T + 2.29e8T^{2} \)
53 \( 1 - 1.49e3iT - 4.18e8T^{2} \)
59 \( 1 - 6.36e3T + 7.14e8T^{2} \)
61 \( 1 - 5.09e3T + 8.44e8T^{2} \)
67 \( 1 + 5.87e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.46e4T + 1.80e9T^{2} \)
73 \( 1 - 3.48e4T + 2.07e9T^{2} \)
79 \( 1 - 3.71e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.77e4T + 3.93e9T^{2} \)
89 \( 1 - 3.84e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.48e5T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.28698395249524346919617629139, −9.447705339432828157875911025565, −8.289931902906358724458397275870, −7.49841765749351610621382937365, −6.78336118493382082982814041080, −5.19451469547382299380565187037, −4.13772504875395329481775228040, −2.95710142783520186780744872280, −2.19787945316370388259260786246, −0.26943015696211501423976511159, 1.46389045330974959423673306162, 2.47126847267134021184357864829, 3.66438278423693814944024523018, 4.92383409132516740409656442202, 5.91869760987054791174428063234, 7.31248159555116277460419578696, 8.218978723649414670695664579749, 8.693822923144566906459156595425, 9.811019127817385571789567937885, 10.49414124441827839278313107916

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