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Properties

Label 2-1776-111.110-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.356584278\)
\(L(\frac12)\) \(\approx\) \(1.356584278\)
\(L(1)\) 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

−9.325054677920461449105578845317, −8.523209355728626700386277236911, −7.965544001346385165724090064048, −7.35181616075604460950650101971, −6.61100799957585136596190739615, −5.16481271529654733240152396222, −4.35580750414592608714486247682, −3.43080518028761430301594679409, −2.90887132766063790590384141885, −1.25810513308472643295802697626, 1.25810513308472643295802697626, 2.90887132766063790590384141885, 3.43080518028761430301594679409, 4.35580750414592608714486247682, 5.16481271529654733240152396222, 6.61100799957585136596190739615, 7.35181616075604460950650101971, 7.965544001346385165724090064048, 8.523209355728626700386277236911, 9.325054677920461449105578845317

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