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Properties

Label 2-38e2-19.18-c2-0-15
Degree $2$
Conductor $1444$
Sign $-0.575 - 0.817i$
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  + 3.41i·3-s − 9.09·5-s − 4.38·7-s − 2.63·9-s + 18.2·11-s − 11.8i·13-s − 31.0i·15-s − 8.65·17-s − 14.9i·21-s + 33.6·23-s + 57.6·25-s + 21.7i·27-s + 49.6i·29-s − 9.75i·31-s + 62.2i·33-s + ⋯
L(s)  = 1  + 1.13i·3-s − 1.81·5-s − 0.626·7-s − 0.292·9-s + 1.65·11-s − 0.912i·13-s − 2.06i·15-s − 0.509·17-s − 0.712i·21-s + 1.46·23-s + 2.30·25-s + 0.804i·27-s + 1.71i·29-s − 0.314i·31-s + 1.88i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.575 - 0.817i$
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: \((2,\ 1444,\ (\ :1),\ -0.575 - 0.817i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.078498014\)
\(L(\frac12)\) \(\approx\) \(1.078498014\)
\(L(2)\) 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 \( 1 - 3.41iT - 9T^{2} \)
5 \( 1 + 9.09T + 25T^{2} \)
7 \( 1 + 4.38T + 49T^{2} \)
11 \( 1 - 18.2T + 121T^{2} \)
13 \( 1 + 11.8iT - 169T^{2} \)
17 \( 1 + 8.65T + 289T^{2} \)
23 \( 1 - 33.6T + 529T^{2} \)
29 \( 1 - 49.6iT - 841T^{2} \)
31 \( 1 + 9.75iT - 961T^{2} \)
37 \( 1 + 44.8iT - 1.36e3T^{2} \)
41 \( 1 - 43.7iT - 1.68e3T^{2} \)
43 \( 1 - 5.45T + 1.84e3T^{2} \)
47 \( 1 - 15.2T + 2.20e3T^{2} \)
53 \( 1 + 16.5iT - 2.80e3T^{2} \)
59 \( 1 + 31.4iT - 3.48e3T^{2} \)
61 \( 1 + 37.3T + 3.72e3T^{2} \)
67 \( 1 + 8.06iT - 4.48e3T^{2} \)
71 \( 1 + 123. iT - 5.04e3T^{2} \)
73 \( 1 + 4.12T + 5.32e3T^{2} \)
79 \( 1 - 77.7iT - 6.24e3T^{2} \)
83 \( 1 - 61.6T + 6.88e3T^{2} \)
89 \( 1 + 33.8iT - 7.92e3T^{2} \)
97 \( 1 - 188. iT - 9.40e3T^{2} \)
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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

−9.291254042657563232541240983018, −9.125792423688083910032050796479, −8.102622301357312758748216967144, −7.15850542562566542849745720732, −6.55400441418926520665117363395, −5.11585612843853114450648529056, −4.39502087676144708715143325156, −3.58639840543684735375292012215, −3.22302981245822281366023028315, −0.939810900782701365963715083193, 0.42503368577258386588701547114, 1.45188257768638788167954893104, 2.93021092756826490493590045332, 3.99700030755753326730923872560, 4.47438387217115703503291137042, 6.18889971572735593977963286681, 7.00711418412824492287170248996, 7.08258469261931282469497721186, 8.204360849056542577144089461855, 8.842173424270564105799938712306

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