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Properties

Label 2-38e2-19.18-c2-0-6
Degree $2$
Conductor $1444$
Sign $-0.422 + 0.906i$
Analytic cond. $39.3461$
Root an. cond. $6.27265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.75i·3-s + 6.82·5-s + 2.75·7-s − 24.0·9-s − 17.6·11-s + 12.8i·13-s + 39.2i·15-s − 19.1·17-s + 15.8i·21-s − 0.789·23-s + 21.6·25-s − 86.6i·27-s − 7.93i·29-s − 17.6i·31-s − 101. i·33-s + ⋯
L(s)  = 1  + 1.91i·3-s + 1.36·5-s + 0.394·7-s − 2.67·9-s − 1.60·11-s + 0.988i·13-s + 2.61i·15-s − 1.12·17-s + 0.755i·21-s − 0.0343·23-s + 0.864·25-s − 3.20i·27-s − 0.273i·29-s − 0.567i·31-s − 3.06i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.422 + 0.906i$
Analytic conductor: \(39.3461\)
Root analytic conductor: \(6.27265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :1),\ -0.422 + 0.906i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8156023878\)
\(L(\frac12)\) \(\approx\) \(0.8156023878\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 - 5.75iT - 9T^{2} \)
5 \( 1 - 6.82T + 25T^{2} \)
7 \( 1 - 2.75T + 49T^{2} \)
11 \( 1 + 17.6T + 121T^{2} \)
13 \( 1 - 12.8iT - 169T^{2} \)
17 \( 1 + 19.1T + 289T^{2} \)
23 \( 1 + 0.789T + 529T^{2} \)
29 \( 1 + 7.93iT - 841T^{2} \)
31 \( 1 + 17.6iT - 961T^{2} \)
37 \( 1 - 1.64iT - 1.36e3T^{2} \)
41 \( 1 + 9.96iT - 1.68e3T^{2} \)
43 \( 1 - 21.3T + 1.84e3T^{2} \)
47 \( 1 - 53.7T + 2.20e3T^{2} \)
53 \( 1 - 42.3iT - 2.80e3T^{2} \)
59 \( 1 - 93.5iT - 3.48e3T^{2} \)
61 \( 1 + 84.2T + 3.72e3T^{2} \)
67 \( 1 + 12.1iT - 4.48e3T^{2} \)
71 \( 1 + 20.8iT - 5.04e3T^{2} \)
73 \( 1 + 92.5T + 5.32e3T^{2} \)
79 \( 1 + 77.2iT - 6.24e3T^{2} \)
83 \( 1 - 35.3T + 6.88e3T^{2} \)
89 \( 1 + 2.37iT - 7.92e3T^{2} \)
97 \( 1 + 149. iT - 9.40e3T^{2} \)
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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

−9.949528373980451078814626951698, −9.145158046319419089661136969739, −8.791984975174030487123371123205, −7.62872712186319309577222610668, −6.18023481659226704007433964675, −5.61529195310733475406193183375, −4.78533630735935336933447382707, −4.24782861691901210310353529806, −2.85203115026153763125155208154, −2.14575403500466928775991394917, 0.19983039305663173201144319080, 1.47436996096126048272898487238, 2.32104149042798382407811174375, 2.89555732440882722057737475574, 5.08017857218362128272035331302, 5.62255137903672547630839392443, 6.36356828923390945457223289577, 7.15491810807346267571280166350, 7.969583199936724390231493559855, 8.473844807274357236722849044014

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