Properties

Label 2-38e2-76.43-c0-0-3
Degree $2$
Conductor $1444$
Sign $-0.529 + 0.848i$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−1.23 − 1.04i)5-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−1.52 − 0.553i)10-s + (−0.107 − 0.608i)13-s + (0.173 − 0.984i)16-s + (−0.580 + 0.211i)17-s − 18-s − 1.61·20-s + (0.280 + 1.59i)25-s + (−0.309 − 0.535i)26-s + (0.580 + 0.211i)29-s + (−0.173 − 0.984i)32-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−1.23 − 1.04i)5-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−1.52 − 0.553i)10-s + (−0.107 − 0.608i)13-s + (0.173 − 0.984i)16-s + (−0.580 + 0.211i)17-s − 18-s − 1.61·20-s + (0.280 + 1.59i)25-s + (−0.309 − 0.535i)26-s + (0.580 + 0.211i)29-s + (−0.173 − 0.984i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.529 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341981942\)
\(L(\frac12)\) \(\approx\) \(1.341981942\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.580 - 0.211i)T + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.61T + T^{2} \)
41 \( 1 + (-0.280 + 1.59i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-1.23 + 1.04i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.409533761398737793214908944936, −8.559148092534384642704337328633, −7.897846845132654088037882208748, −6.93501522156106692906709138561, −5.88855857315006309167638803147, −5.13946081901521000304474046282, −4.28940816588751869370630059941, −3.57423955442126152535036262086, −2.51360768531416406287295346914, −0.801908867278651279463621912622, 2.43922853860226976871691112071, 3.11369423793324844311584924725, 4.10265195795682511731186264309, 4.80626947083717748812947982612, 6.04390269995690363425303168658, 6.65564419732358591193307074789, 7.51911956886964515148378333656, 8.041334260872681272509252314773, 8.930283851372758145363040131434, 10.29452315147806604102814674665

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