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Properties

Label 2-38e2-19.18-c2-0-17
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  + 1.77i·3-s + 5.00·5-s + 3.79·7-s + 5.83·9-s − 11.3·11-s + 8.41i·13-s + 8.90i·15-s − 22.6·17-s + 6.74i·21-s − 17.7·23-s + 0.0808·25-s + 26.3i·27-s + 21.3i·29-s + 48.7i·31-s − 20.2i·33-s + ⋯
L(s)  = 1  + 0.592i·3-s + 1.00·5-s + 0.541·7-s + 0.648·9-s − 1.03·11-s + 0.647i·13-s + 0.593i·15-s − 1.33·17-s + 0.321i·21-s − 0.771·23-s + 0.00323·25-s + 0.977i·27-s + 0.734i·29-s + 1.57i·31-s − 0.613i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.828625521\)
\(L(\frac12)\) \(\approx\) \(1.828625521\)
\(L(2)\) 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 \( 1 - 1.77iT - 9T^{2} \)
5 \( 1 - 5.00T + 25T^{2} \)
7 \( 1 - 3.79T + 49T^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 - 8.41iT - 169T^{2} \)
17 \( 1 + 22.6T + 289T^{2} \)
23 \( 1 + 17.7T + 529T^{2} \)
29 \( 1 - 21.3iT - 841T^{2} \)
31 \( 1 - 48.7iT - 961T^{2} \)
37 \( 1 - 32.1iT - 1.36e3T^{2} \)
41 \( 1 - 27.7iT - 1.68e3T^{2} \)
43 \( 1 + 45.9T + 1.84e3T^{2} \)
47 \( 1 - 35.7T + 2.20e3T^{2} \)
53 \( 1 + 65.2iT - 2.80e3T^{2} \)
59 \( 1 + 19.4iT - 3.48e3T^{2} \)
61 \( 1 - 88.8T + 3.72e3T^{2} \)
67 \( 1 - 54.8iT - 4.48e3T^{2} \)
71 \( 1 - 80.1iT - 5.04e3T^{2} \)
73 \( 1 - 108.T + 5.32e3T^{2} \)
79 \( 1 - 101. iT - 6.24e3T^{2} \)
83 \( 1 - 75.6T + 6.88e3T^{2} \)
89 \( 1 + 164. iT - 7.92e3T^{2} \)
97 \( 1 + 75.3iT - 9.40e3T^{2} \)
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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

−9.828681027366208015432840315825, −8.872054541844611213590727986621, −8.204919758202657612858593713228, −7.04271415540403129001301897862, −6.40484544999908603822817496796, −5.17124203807080930482401894146, −4.81724612141193700018390051875, −3.70416591897447731103842418401, −2.36412921292285251783583793683, −1.55538774662488725498058306865, 0.46486125486015718496506845376, 1.97594744961052234127433669131, 2.35546653326640119139990569523, 3.99974155539500002950911443559, 4.98128722065508027690009081052, 5.84896227679823216727273450663, 6.52022697956996884293711335990, 7.62471088214950080166461068436, 8.005205369725834453649667308973, 9.115132621355109514997335382360

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