Properties

Label 2-1776-111.110-c0-0-1
Degree $2$
Conductor $1776$
Sign $1$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s − 2·21-s − 25-s − 27-s + 37-s + 3·49-s + 2·63-s + 2·67-s − 2·73-s + 75-s + 81-s − 111-s + ⋯
L(s)  = 1  − 3-s + 2·7-s + 9-s − 2·21-s − 25-s − 27-s + 37-s + 3·49-s + 2·63-s + 2·67-s − 2·73-s + 75-s + 81-s − 111-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: yes
Arithmetic: yes
Character: $\chi_{1776} (1553, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050052551\)
\(L(\frac12)\) \(\approx\) \(1.050052551\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( ( 1 - T )^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.607802282887666890290522230699, −8.544691620823811235250958133716, −7.82182909815563501771744016286, −7.21518932728387963466040801750, −6.10107227607268113720615972869, −5.37241650071437557940613373125, −4.67740826158460570736013309643, −3.97097573600309252851361150463, −2.19834462836226889317934222558, −1.22840974389690426951788623925, 1.22840974389690426951788623925, 2.19834462836226889317934222558, 3.97097573600309252851361150463, 4.67740826158460570736013309643, 5.37241650071437557940613373125, 6.10107227607268113720615972869, 7.21518932728387963466040801750, 7.82182909815563501771744016286, 8.544691620823811235250958133716, 9.607802282887666890290522230699

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