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Properties

Label 2-38e2-76.63-c0-0-3
Degree $2$
Conductor $1444$
Sign $-0.570 + 0.821i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.280 − 1.59i)5-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−1.23 − 1.04i)10-s + (−0.580 + 0.211i)13-s + (−0.939 − 0.342i)16-s + (0.473 − 0.397i)17-s + 18-s − 1.61·20-s + (−1.52 + 0.553i)25-s + (−0.309 + 0.535i)26-s + (0.473 + 0.397i)29-s + (−0.939 + 0.342i)32-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.280 − 1.59i)5-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−1.23 − 1.04i)10-s + (−0.580 + 0.211i)13-s + (−0.939 − 0.342i)16-s + (0.473 − 0.397i)17-s + 18-s − 1.61·20-s + (−1.52 + 0.553i)25-s + (−0.309 + 0.535i)26-s + (0.473 + 0.397i)29-s + (−0.939 + 0.342i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.566772710\)
\(L(\frac12)\) \(\approx\) \(1.566772710\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.552243763307015281469512921212, −8.836608172867077561131895629229, −7.84307476688154384417517618182, −6.99741118818289664184271052550, −5.77296367173843587425826207317, −4.86472471032111823691560929604, −4.62697635339623561951180133665, −3.53065820350263798488780819688, −2.13188352626739898071159879540, −1.08984087481933387496895294745, 2.29712804652497751872714700174, 3.32619257106399550141421428257, 3.90257074081544830939803420707, 5.02877578141790249654889581366, 6.15625964276088019952619960882, 6.69017110652789799069692509025, 7.40288930677842120329273766316, 7.952679256644695695346662254844, 9.165170667301254790701049406930, 10.13828709429759182011775678909

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