Properties

Label 2-1944-216.67-c0-0-3
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.5 − 0.866i)8-s + (1.17 − 0.984i)11-s + (0.766 − 0.642i)16-s + (0.939 − 1.62i)17-s + (−0.173 − 0.300i)19-s + (1.17 + 0.984i)22-s + (0.173 + 0.984i)25-s + (0.766 + 0.642i)32-s + (1.76 + 0.642i)34-s + (0.266 − 0.223i)38-s + (0.0603 − 0.342i)41-s + (−1.43 + 1.20i)43-s + (−0.766 + 1.32i)44-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.5 − 0.866i)8-s + (1.17 − 0.984i)11-s + (0.766 − 0.642i)16-s + (0.939 − 1.62i)17-s + (−0.173 − 0.300i)19-s + (1.17 + 0.984i)22-s + (0.173 + 0.984i)25-s + (0.766 + 0.642i)32-s + (1.76 + 0.642i)34-s + (0.266 − 0.223i)38-s + (0.0603 − 0.342i)41-s + (−1.43 + 1.20i)43-s + (−0.766 + 1.32i)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: $\chi_{1944} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204916089\)
\(L(\frac12)\) \(\approx\) \(1.204916089\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.299595254252309832904547843738, −8.667706698449084858542184609098, −7.80017948548131073751146776421, −7.05546286392487499445781236814, −6.37563602507622255187596912409, −5.52325684391100631712637779192, −4.81770258966898918946966693137, −3.73347602510536179227797238339, −3.01224616975296691773175399169, −1.02700077572391596750237886066, 1.35981202647309825896168753469, 2.20895382781020691492426975177, 3.58947708986422348532754412386, 4.07270392229916274098671787366, 5.05186480967533061509192256783, 6.01409961405991888550015379226, 6.79818309452422434409409088118, 8.004269583820269675503443290453, 8.631192401909108119143266667319, 9.471067256635339345762760029218

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