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Properties

Label 2-38e2-19.18-c2-0-19
Degree $2$
Conductor $1444$
Sign $-0.909 - 0.416i$
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  + 4.56i·3-s − 0.176·5-s + 5.77·7-s − 11.8·9-s + 15.5·11-s − 5.51i·13-s − 0.803i·15-s − 30.4·17-s + 26.3i·21-s + 21.5·23-s − 24.9·25-s − 12.9i·27-s + 4.05i·29-s + 49.3i·31-s + 70.8i·33-s + ⋯
L(s)  = 1  + 1.52i·3-s − 0.0352·5-s + 0.825·7-s − 1.31·9-s + 1.41·11-s − 0.424i·13-s − 0.0535i·15-s − 1.79·17-s + 1.25i·21-s + 0.937·23-s − 0.998·25-s − 0.479i·27-s + 0.139i·29-s + 1.59i·31-s + 2.14i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.909 - 0.416i$
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: \((2,\ 1444,\ (\ :1),\ -0.909 - 0.416i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.805181856\)
\(L(\frac12)\) \(\approx\) \(1.805181856\)
\(L(2)\) 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 \( 1 - 4.56iT - 9T^{2} \)
5 \( 1 + 0.176T + 25T^{2} \)
7 \( 1 - 5.77T + 49T^{2} \)
11 \( 1 - 15.5T + 121T^{2} \)
13 \( 1 + 5.51iT - 169T^{2} \)
17 \( 1 + 30.4T + 289T^{2} \)
23 \( 1 - 21.5T + 529T^{2} \)
29 \( 1 - 4.05iT - 841T^{2} \)
31 \( 1 - 49.3iT - 961T^{2} \)
37 \( 1 - 62.3iT - 1.36e3T^{2} \)
41 \( 1 + 31.1iT - 1.68e3T^{2} \)
43 \( 1 - 73.2T + 1.84e3T^{2} \)
47 \( 1 + 44.9T + 2.20e3T^{2} \)
53 \( 1 - 69.5iT - 2.80e3T^{2} \)
59 \( 1 - 14.8iT - 3.48e3T^{2} \)
61 \( 1 + 30.5T + 3.72e3T^{2} \)
67 \( 1 - 99.0iT - 4.48e3T^{2} \)
71 \( 1 - 104. iT - 5.04e3T^{2} \)
73 \( 1 + 0.936T + 5.32e3T^{2} \)
79 \( 1 - 20.1iT - 6.24e3T^{2} \)
83 \( 1 - 39.7T + 6.88e3T^{2} \)
89 \( 1 - 10.4iT - 7.92e3T^{2} \)
97 \( 1 - 26.7iT - 9.40e3T^{2} \)
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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

−9.640180376707382176614133558529, −8.883827970797593668563794380754, −8.538234848495673712983948453833, −7.22379799872023822266633652125, −6.33518096237312593380686707227, −5.25099566917314616150209057258, −4.51162442097269795630957196279, −3.97748905229507607823578763420, −2.88346124380086939016281968433, −1.41359586096209334010364093610, 0.51015171270555596927478182982, 1.71496751839760504172858046058, 2.25314495039872099713343807227, 3.87880956552808760982213431103, 4.74132528863876693996340951986, 6.11839253864112302413068246129, 6.52233093719266739009318330005, 7.40325727035404649899179702597, 7.979872820306127945203141865124, 8.959751812427271389652701625215

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