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Properties

Label 2-38e2-19.18-c2-0-12
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  + 0.174i·3-s − 0.292·5-s − 5.22·7-s + 8.96·9-s + 3.31·11-s + 12.1i·13-s − 0.0510i·15-s − 7.62·17-s − 0.910i·21-s + 8.91·23-s − 24.9·25-s + 3.13i·27-s − 3.24i·29-s + 15.5i·31-s + 0.578i·33-s + ⋯
L(s)  = 1  + 0.0581i·3-s − 0.0585·5-s − 0.746·7-s + 0.996·9-s + 0.301·11-s + 0.934i·13-s − 0.00340i·15-s − 0.448·17-s − 0.0433i·21-s + 0.387·23-s − 0.996·25-s + 0.116i·27-s − 0.111i·29-s + 0.502i·31-s + 0.0175i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.151158033\)
\(L(\frac12)\) \(\approx\) \(1.151158033\)
\(L(2)\) 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 \( 1 - 0.174iT - 9T^{2} \)
5 \( 1 + 0.292T + 25T^{2} \)
7 \( 1 + 5.22T + 49T^{2} \)
11 \( 1 - 3.31T + 121T^{2} \)
13 \( 1 - 12.1iT - 169T^{2} \)
17 \( 1 + 7.62T + 289T^{2} \)
23 \( 1 - 8.91T + 529T^{2} \)
29 \( 1 + 3.24iT - 841T^{2} \)
31 \( 1 - 15.5iT - 961T^{2} \)
37 \( 1 + 35.5iT - 1.36e3T^{2} \)
41 \( 1 - 35.0iT - 1.68e3T^{2} \)
43 \( 1 - 45.0T + 1.84e3T^{2} \)
47 \( 1 + 56.0T + 2.20e3T^{2} \)
53 \( 1 + 53.1iT - 2.80e3T^{2} \)
59 \( 1 - 95.4iT - 3.48e3T^{2} \)
61 \( 1 - 64.9T + 3.72e3T^{2} \)
67 \( 1 - 37.3iT - 4.48e3T^{2} \)
71 \( 1 - 86.4iT - 5.04e3T^{2} \)
73 \( 1 + 23.5T + 5.32e3T^{2} \)
79 \( 1 - 120. iT - 6.24e3T^{2} \)
83 \( 1 + 106.T + 6.88e3T^{2} \)
89 \( 1 - 112. iT - 7.92e3T^{2} \)
97 \( 1 - 43.1iT - 9.40e3T^{2} \)
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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

−9.598136034396204696084296714566, −8.984576878455392567835063170281, −7.953587359993214500725724334050, −6.95049388339270264913450203384, −6.58670722798902514505571012282, −5.47500942143688464644734550881, −4.33143772209371988110057475728, −3.77410407973349730109216743566, −2.46588142821101269623333583713, −1.28985864472587557699362933565, 0.33279310939787509026217274453, 1.69705372366602763026848071678, 2.99245368568805414529746448303, 3.88898648501947056580511221741, 4.81547654735343061719204064659, 5.90480716983058007322746611065, 6.64435636764857798617365858884, 7.44520072481399430342367702853, 8.201295332873133272535339612840, 9.255514604970630067430073767946

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