Properties

Label 2-384-4.3-c2-0-5
Degree $2$
Conductor $384$
Sign $1$
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  − 1.73i·3-s − 8.29·5-s + 8.55i·7-s − 2.99·9-s − 13.7i·11-s + 17.0·13-s + 14.3i·15-s + 20.3·17-s + 20.4i·19-s + 14.8·21-s − 5.51i·23-s + 43.7·25-s + 5.19i·27-s + 41.0·29-s + 22.2i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.65·5-s + 1.22i·7-s − 0.333·9-s − 1.25i·11-s + 1.31·13-s + 0.957i·15-s + 1.19·17-s + 1.07i·19-s + 0.705·21-s − 0.239i·23-s + 1.75·25-s + 0.192i·27-s + 1.41·29-s + 0.719i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24910\)
\(L(\frac12)\) \(\approx\) \(1.24910\)
\(L(2)\) 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 \( 1 + 1.73iT \)
good5 \( 1 + 8.29T + 25T^{2} \)
7 \( 1 - 8.55iT - 49T^{2} \)
11 \( 1 + 13.7iT - 121T^{2} \)
13 \( 1 - 17.0T + 169T^{2} \)
17 \( 1 - 20.3T + 289T^{2} \)
19 \( 1 - 20.4iT - 361T^{2} \)
23 \( 1 + 5.51iT - 529T^{2} \)
29 \( 1 - 41.0T + 841T^{2} \)
31 \( 1 - 22.2iT - 961T^{2} \)
37 \( 1 + 11.6T + 1.36e3T^{2} \)
41 \( 1 - 35.9T + 1.68e3T^{2} \)
43 \( 1 + 66.8iT - 1.84e3T^{2} \)
47 \( 1 - 19.9iT - 2.20e3T^{2} \)
53 \( 1 - 17.6T + 2.80e3T^{2} \)
59 \( 1 - 62.1iT - 3.48e3T^{2} \)
61 \( 1 + 47.4T + 3.72e3T^{2} \)
67 \( 1 - 74.8iT - 4.48e3T^{2} \)
71 \( 1 - 16.9iT - 5.04e3T^{2} \)
73 \( 1 - 101.T + 5.32e3T^{2} \)
79 \( 1 - 0.879iT - 6.24e3T^{2} \)
83 \( 1 - 23.2iT - 6.88e3T^{2} \)
89 \( 1 + 16.3T + 7.92e3T^{2} \)
97 \( 1 - 188.T + 9.40e3T^{2} \)
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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

−11.36270381033261823096309296640, −10.49278451700455362756112022515, −8.682213756837718503021301386817, −8.476126235338134498221207295951, −7.59395213425854746876390359595, −6.28822593725991666132698449187, −5.48371383370861328287524314874, −3.83150006022524197812799067257, −3.00992939595715092188120593113, −0.972446993435436141898898757880, 0.813620279231900163503385192624, 3.28924334526662370943616769679, 4.11570379484068911403684601267, 4.80384800356699266333610342673, 6.57299493016544889836335495146, 7.55489294554280179308776460062, 8.119338562615278897667758140615, 9.359980794208551611941991769140, 10.39309411472889959978563382088, 11.07291103774402381517781051313

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