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Properties

Label 2-1776-111.104-c0-0-0
Degree $2$
Conductor $1776$
Sign $0.507 - 0.861i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 + 0.342i)3-s + (0.266 + 1.50i)7-s + (0.766 + 0.642i)9-s + (−0.826 − 0.984i)13-s + (−0.592 + 1.62i)19-s + (−0.266 + 1.50i)21-s + (0.939 − 0.342i)25-s + (0.500 + 0.866i)27-s − 1.96i·31-s + (−0.5 + 0.866i)37-s + (−0.439 − 1.20i)39-s − 0.684i·43-s + (−1.26 + 0.460i)49-s + (−1.11 + 1.32i)57-s + (−0.766 + 1.32i)63-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.266 + 1.50i)7-s + (0.766 + 0.642i)9-s + (−0.826 − 0.984i)13-s + (−0.592 + 1.62i)19-s + (−0.266 + 1.50i)21-s + (0.939 − 0.342i)25-s + (0.500 + 0.866i)27-s − 1.96i·31-s + (−0.5 + 0.866i)37-s + (−0.439 − 1.20i)39-s − 0.684i·43-s + (−1.26 + 0.460i)49-s + (−1.11 + 1.32i)57-s + (−0.766 + 1.32i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.558374704\)
\(L(\frac12)\) \(\approx\) \(1.558374704\)
\(L(1)\) 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 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + 1.96iT - T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + 0.684iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 - 1.87T + T^{2} \)
79 \( 1 + (0.673 - 0.118i)T + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.579783275277084525453842908794, −8.727754894213799910160572087542, −8.162355605260806714783724265851, −7.60014859216775822921555071887, −6.31688197375608217748888654307, −5.47739507059147072884058942662, −4.71887358114095515500666920235, −3.59292247758501392538775956292, −2.64517767107957785518444345513, −1.94591137060295496723135560506, 1.15983133025967663996800659770, 2.35447127861083990637006986689, 3.40073313715852699516434220412, 4.36167785860652992537783150108, 4.92772017003713008784337022904, 6.73925285281816429331006093000, 6.95428007212943497773386750937, 7.62996491955644043580192803797, 8.684398910717749315356016182794, 9.153471253248624022659286783828

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