Properties

Label 2-1944-1.1-c1-0-11
Degree $2$
Conductor $1944$
Sign $1$
Analytic cond. $15.5229$
Root an. cond. $3.93991$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.732·5-s + 3.46·7-s + 6.19·11-s − 2.46·13-s + 0.732·17-s − 4.46·19-s + 8.73·23-s − 4.46·25-s + 2.53·29-s + 2.46·31-s − 2.53·35-s + 2.53·37-s + 9.46·41-s − 5.92·43-s − 6.92·47-s + 4.99·49-s − 7.66·53-s − 4.53·55-s + 1.80·59-s − 1.53·61-s + 1.80·65-s − 3.53·67-s − 6.19·71-s + 11.3·73-s + 21.4·77-s + 13.9·79-s + 12.7·83-s + ⋯
L(s)  = 1  − 0.327·5-s + 1.30·7-s + 1.86·11-s − 0.683·13-s + 0.177·17-s − 1.02·19-s + 1.82·23-s − 0.892·25-s + 0.470·29-s + 0.442·31-s − 0.428·35-s + 0.416·37-s + 1.47·41-s − 0.904·43-s − 1.01·47-s + 0.714·49-s − 1.05·53-s − 0.611·55-s + 0.234·59-s − 0.196·61-s + 0.223·65-s − 0.431·67-s − 0.735·71-s + 1.33·73-s + 2.44·77-s + 1.56·79-s + 1.39·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $1$
Analytic conductor: \(15.5229\)
Root analytic conductor: \(3.93991\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.130877280\)
\(L(\frac12)\) \(\approx\) \(2.130877280\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 0.732T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 6.19T + 11T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 - 0.732T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 - 8.73T + 23T^{2} \)
29 \( 1 - 2.53T + 29T^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 + 5.92T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 7.66T + 53T^{2} \)
59 \( 1 - 1.80T + 59T^{2} \)
61 \( 1 + 1.53T + 61T^{2} \)
67 \( 1 + 3.53T + 67T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−9.124704504313158088827163582953, −8.405124705579119643494587238445, −7.69105210399773947620026499265, −6.84603047324666270619531929129, −6.12377124754291391270440856773, −4.86535208042367346346961294623, −4.44066622410118067222608917066, −3.41541263484786576421952861381, −2.07019401611395465061936444601, −1.06713102258613753412883865381, 1.06713102258613753412883865381, 2.07019401611395465061936444601, 3.41541263484786576421952861381, 4.44066622410118067222608917066, 4.86535208042367346346961294623, 6.12377124754291391270440856773, 6.84603047324666270619531929129, 7.69105210399773947620026499265, 8.405124705579119643494587238445, 9.124704504313158088827163582953

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