Properties

Label 2-1776-111.110-c0-0-3
Degree $2$
Conductor $1776$
Sign $1$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.41·5-s + 9-s + 1.41·15-s − 1.41·17-s − 1.41·23-s + 1.00·25-s + 27-s − 1.41·29-s − 37-s + 1.41·45-s − 49-s − 1.41·51-s + 1.41·59-s − 1.41·69-s + 1.00·75-s + 81-s − 2.00·85-s − 1.41·87-s + 1.41·89-s − 111-s + 1.41·113-s − 2.00·115-s + ⋯
L(s)  = 1  + 3-s + 1.41·5-s + 9-s + 1.41·15-s − 1.41·17-s − 1.41·23-s + 1.00·25-s + 27-s − 1.41·29-s − 37-s + 1.41·45-s − 49-s − 1.41·51-s + 1.41·59-s − 1.41·69-s + 1.00·75-s + 81-s − 2.00·85-s − 1.41·87-s + 1.41·89-s − 111-s + 1.41·113-s − 2.00·115-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1553, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.921574238\)
\(L(\frac12)\) \(\approx\) \(1.921574238\)
\(L(1)\) 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 \( 1 - T \)
37 \( 1 + T \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.426173882223015863852285668220, −8.844391903596422675846563968073, −8.060460267554353381103231348907, −7.07253664058948919393827658611, −6.35329881595573979449596344814, −5.48856217915871096384010677831, −4.45666397988292942789989544860, −3.49056794757306326801487732408, −2.24946433102945990229629677595, −1.82859637069686783306816551439, 1.82859637069686783306816551439, 2.24946433102945990229629677595, 3.49056794757306326801487732408, 4.45666397988292942789989544860, 5.48856217915871096384010677831, 6.35329881595573979449596344814, 7.07253664058948919393827658611, 8.060460267554353381103231348907, 8.844391903596422675846563968073, 9.426173882223015863852285668220

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