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Properties

Label 2-1776-444.131-c0-0-0
Degree $2$
Conductor $1776$
Sign $0.0320 - 0.999i$
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 + (−1.20 + 1.43i)7-s + (−0.939 − 0.342i)9-s + (−0.766 + 1.64i)13-s + (−1.58 + 1.10i)19-s + (1.20 + 1.43i)21-s + (0.984 − 0.173i)25-s + (−0.5 + 0.866i)27-s + (0.123 − 0.123i)31-s + (0.5 + 0.866i)37-s + (1.48 + 1.03i)39-s + (−1.15 − 1.15i)43-s + (−0.439 − 2.49i)49-s + (0.816 + 1.75i)57-s + (−1.28 − 0.597i)61-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−1.20 + 1.43i)7-s + (−0.939 − 0.342i)9-s + (−0.766 + 1.64i)13-s + (−1.58 + 1.10i)19-s + (1.20 + 1.43i)21-s + (0.984 − 0.173i)25-s + (−0.5 + 0.866i)27-s + (0.123 − 0.123i)31-s + (0.5 + 0.866i)37-s + (1.48 + 1.03i)39-s + (−1.15 − 1.15i)43-s + (−0.439 − 2.49i)49-s + (0.816 + 1.75i)57-s + (−1.28 − 0.597i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.0320 - 0.999i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.0320 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5966369677\)
\(L(\frac12)\) \(\approx\) \(0.5966369677\)
\(L(1)\) 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 \( 1 + (-0.984 + 0.173i)T^{2} \)
7 \( 1 + (1.20 - 1.43i)T + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 - 1.64i)T + (-0.642 - 0.766i)T^{2} \)
17 \( 1 + (-0.642 + 0.766i)T^{2} \)
19 \( 1 + (1.58 - 1.10i)T + (0.342 - 0.939i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (-0.123 + 0.123i)T - iT^{2} \)
41 \( 1 + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (1.15 + 1.15i)T + iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.984 - 0.173i)T^{2} \)
61 \( 1 + (1.28 + 0.597i)T + (0.642 + 0.766i)T^{2} \)
67 \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.939 - 0.342i)T^{2} \)
73 \( 1 - 0.347iT - T^{2} \)
79 \( 1 + (0.0999 + 1.14i)T + (-0.984 + 0.173i)T^{2} \)
83 \( 1 + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (-0.984 - 0.173i)T^{2} \)
97 \( 1 + (-0.296 - 1.10i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.457070175640164776783858566628, −8.798601666236562155274597032525, −8.301290681748740499740888034354, −7.07229283763600709311521203828, −6.46837657877193108323020227355, −6.02062672445843639277812663244, −4.88124306656530914627085927365, −3.60175198219902621680082860898, −2.52744136370217502305991839895, −1.91855629349392536293080238544, 0.40520515076362791431612246528, 2.71803322619783900649750196460, 3.33170750399403947822794336287, 4.30571832944959234574066032199, 4.97710178256610577871298668792, 6.10811141386518599648046291448, 6.89689474912579342157299944968, 7.74691188006513350064818922652, 8.595073301031212497777013813376, 9.533409908283389976407355114993

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