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Properties

Label 2-1776-111.41-c0-0-0
Degree $2$
Conductor $1776$
Sign $0.568 - 0.822i$
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  + (−0.173 − 0.984i)3-s + (−1.43 + 1.20i)7-s + (−0.939 + 0.342i)9-s + (−0.233 + 0.642i)13-s + (1.70 − 0.300i)19-s + (1.43 + 1.20i)21-s + (−0.173 + 0.984i)25-s + (0.5 + 0.866i)27-s + 1.28i·31-s + (−0.5 + 0.866i)37-s + (0.673 + 0.118i)39-s + 1.96i·43-s + (0.439 − 2.49i)49-s + (−0.592 − 1.62i)57-s + (0.939 − 1.62i)63-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−1.43 + 1.20i)7-s + (−0.939 + 0.342i)9-s + (−0.233 + 0.642i)13-s + (1.70 − 0.300i)19-s + (1.43 + 1.20i)21-s + (−0.173 + 0.984i)25-s + (0.5 + 0.866i)27-s + 1.28i·31-s + (−0.5 + 0.866i)37-s + (0.673 + 0.118i)39-s + 1.96i·43-s + (0.439 − 2.49i)49-s + (−0.592 − 1.62i)57-s + (0.939 − 1.62i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.568 - 0.822i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6937349067\)
\(L(\frac12)\) \(\approx\) \(0.6937349067\)
\(L(1)\) 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 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T^{2} \)
19 \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - 1.28iT - T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 - 1.96iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + 0.347T + T^{2} \)
79 \( 1 + (1.26 + 1.50i)T + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.346663904674486812575201044978, −8.991716511927994191848489591788, −7.946133213146817371134969990651, −7.04760471456088917646327917328, −6.51148515988391085215698303890, −5.70285224642211111692791511098, −5.01273825546687187834992924991, −3.28540791601475972579553731640, −2.79785458701107913779754346510, −1.49912772638998169600112054499, 0.54294505215692146883286300459, 2.75931047402259033012746961469, 3.60346629829847162360796310563, 4.15478183300101873380372621459, 5.35462447914336679302333057290, 6.01002169391256970834039747464, 7.01042393444117663879049195539, 7.68325962249162112869554134570, 8.781914199023044141953123091471, 9.714271111875983449379123740971

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