Properties

Label 2-38e2-76.55-c0-0-0
Degree $2$
Conductor $1444$
Sign $0.782 - 0.622i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.580 + 0.211i)5-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.107 + 0.608i)10-s + (−1.23 + 1.04i)13-s + (0.766 + 0.642i)16-s + (−0.280 + 1.59i)17-s + 0.999·18-s + 0.618·20-s + (−0.473 + 0.397i)25-s + (0.809 + 1.40i)26-s + (−0.280 − 1.59i)29-s + (0.766 − 0.642i)32-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.580 + 0.211i)5-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.107 + 0.608i)10-s + (−1.23 + 1.04i)13-s + (0.766 + 0.642i)16-s + (−0.280 + 1.59i)17-s + 0.999·18-s + 0.618·20-s + (−0.473 + 0.397i)25-s + (0.809 + 1.40i)26-s + (−0.280 − 1.59i)29-s + (0.766 − 0.642i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (967, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.782 - 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6699186570\)
\(L(\frac12)\) \(\approx\) \(0.6699186570\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.580 + 0.211i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.904725521983053871702308878307, −9.271580853591602288732944228618, −8.143632433862134869418297566134, −7.69883343414819464135310578099, −6.48132483770986526735963228861, −5.43307049901973074638091711580, −4.40646953784592702974863273673, −3.97159968280012428603815074613, −2.57948173831627851045723270462, −1.79808480386251094680648567362, 0.51461945637253604653206781229, 2.86051692980694759961435183125, 3.78495361417254568026315668123, 4.81278174172725645735718563019, 5.39258421306766842729198586702, 6.53666965389014589385047943860, 7.24787034027909224754028033921, 7.81188774167622072915727254168, 8.751441604749329810858438474645, 9.483224284241307686461651207606

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