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Properties

Label 2-384-3.2-c4-0-56
Degree $2$
Conductor $384$
Sign $0.0128 + 0.999i$
Analytic cond. $39.6940$
Root an. cond. $6.30032$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.99 − 0.115i)3-s − 25.0i·5-s + 59.0·7-s + (80.9 − 2.07i)9-s − 219. i·11-s + 0.112·13-s + (−2.88 − 225. i)15-s − 160. i·17-s − 592.·19-s + (531. − 6.80i)21-s − 11.6i·23-s − 1.46·25-s + (728. − 27.9i)27-s + 1.26e3i·29-s + 217.·31-s + ⋯
L(s)  = 1  + (0.999 − 0.0128i)3-s − 1.00i·5-s + 1.20·7-s + (0.999 − 0.0256i)9-s − 1.81i·11-s + 0.000667·13-s + (−0.0128 − 1.00i)15-s − 0.556i·17-s − 1.64·19-s + (1.20 − 0.0154i)21-s − 0.0219i·23-s − 0.00233·25-s + (0.999 − 0.0384i)27-s + 1.49i·29-s + 0.225·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(39.6940\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.209271424\)
\(L(\frac12)\) \(\approx\) \(3.209271424\)
\(L(3)\) 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 \( 1 + (-8.99 + 0.115i)T \)
good5 \( 1 + 25.0iT - 625T^{2} \)
7 \( 1 - 59.0T + 2.40e3T^{2} \)
11 \( 1 + 219. iT - 1.46e4T^{2} \)
13 \( 1 - 0.112T + 2.85e4T^{2} \)
17 \( 1 + 160. iT - 8.35e4T^{2} \)
19 \( 1 + 592.T + 1.30e5T^{2} \)
23 \( 1 + 11.6iT - 2.79e5T^{2} \)
29 \( 1 - 1.26e3iT - 7.07e5T^{2} \)
31 \( 1 - 217.T + 9.23e5T^{2} \)
37 \( 1 + 1.94e3T + 1.87e6T^{2} \)
41 \( 1 + 1.16e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.08e3T + 3.41e6T^{2} \)
47 \( 1 - 3.11e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.79e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.25e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.33e3T + 1.38e7T^{2} \)
67 \( 1 + 2.64e3T + 2.01e7T^{2} \)
71 \( 1 - 6.53e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.38e3T + 2.83e7T^{2} \)
79 \( 1 - 4.15e3T + 3.89e7T^{2} \)
83 \( 1 - 6.49e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.68e3iT - 6.27e7T^{2} \)
97 \( 1 - 274.T + 8.85e7T^{2} \)
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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

−10.58134524993464202456958090212, −9.161779561827993003295975764118, −8.468739725697423113140642169632, −8.227598565694980386399521960145, −6.83996674588191146765351354924, −5.39451488856416892584480479146, −4.51482761212139832623862276348, −3.34729707762074334475176338161, −1.91961879676099284022111173040, −0.795232335723055284679400218751, 1.79640143623422172305798497065, 2.42449509528252762303892876702, 3.98937764662905178180674152576, 4.73295728583916418734988825281, 6.47577658447328648997497381224, 7.35261413972147517959923194792, 8.062126949000649847618583505797, 9.020868821608007212927146024520, 10.21693892401617755178107334241, 10.60759872260143193567572745765

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