Properties

Label 2-1944-216.115-c0-0-1
Degree $2$
Conductor $1944$
Sign $0.727 + 0.686i$
Analytic cond. $0.970182$
Root an. cond. $0.984978$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.500 + 0.866i)8-s + (0.0603 − 0.342i)11-s + (0.173 − 0.984i)16-s + (−0.766 − 1.32i)17-s + (0.939 − 1.62i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)25-s + (0.173 + 0.984i)32-s + (1.17 + 0.984i)34-s + (−0.326 + 1.85i)38-s + (1.76 + 0.642i)41-s + (0.266 − 1.50i)43-s + (−0.173 − 0.300i)44-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.500 + 0.866i)8-s + (0.0603 − 0.342i)11-s + (0.173 − 0.984i)16-s + (−0.766 − 1.32i)17-s + (0.939 − 1.62i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)25-s + (0.173 + 0.984i)32-s + (1.17 + 0.984i)34-s + (−0.326 + 1.85i)38-s + (1.76 + 0.642i)41-s + (0.266 − 1.50i)43-s + (−0.173 − 0.300i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6894936804\)
\(L(\frac12)\) \(\approx\) \(0.6894936804\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.287754009308951798588358003772, −8.665547324002765287191858763053, −7.57938401035405872648975529956, −7.19199280039092111793815717043, −6.29000777858424247564983748595, −5.42231883886997333128942299133, −4.58815469625941458868910049291, −3.10424445265575968068013332402, −2.23806237755771189090842454127, −0.72109630646768855216018339324, 1.40741937195249149060067017829, 2.34076913402442625828128638181, 3.57093019607482095583095089548, 4.28360725707797285438794421648, 5.80551543510447662452795523648, 6.32775618109573510465973524514, 7.48960917540987274172568766491, 7.900166228812693776623074321368, 8.763905773921337214143773541337, 9.491324549225981955366593136932

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