Properties

Label 2-38e2-19.3-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.984 - 0.175i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.483 + 1.32i)3-s + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)7-s + (−0.766 + 0.642i)9-s + (0.5 − 0.866i)11-s + (1.39 − 0.245i)15-s + (0.766 + 0.642i)17-s + (0.909 − 1.08i)21-s + (0.909 + 1.08i)29-s + (1.39 + 0.245i)33-s + (−0.939 + 0.342i)35-s − 1.41i·37-s + (−0.483 − 1.32i)41-s + (−0.173 + 0.984i)43-s + (0.500 + 0.866i)45-s + ⋯
L(s)  = 1  + (0.483 + 1.32i)3-s + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)7-s + (−0.766 + 0.642i)9-s + (0.5 − 0.866i)11-s + (1.39 − 0.245i)15-s + (0.766 + 0.642i)17-s + (0.909 − 1.08i)21-s + (0.909 + 1.08i)29-s + (1.39 + 0.245i)33-s + (−0.939 + 0.342i)35-s − 1.41i·37-s + (−0.483 − 1.32i)41-s + (−0.173 + 0.984i)43-s + (0.500 + 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.984 - 0.175i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310357577\)
\(L(\frac12)\) \(\approx\) \(1.310357577\)
\(L(1)\) 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 \( 1 + (-0.483 - 1.32i)T + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.909 - 1.08i)T + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + (0.483 + 1.32i)T + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.909 - 1.08i)T + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (1.39 + 0.245i)T + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−9.682495364040144313230910102622, −8.953839568047352857084528029676, −8.552954720792161061556540808477, −7.47804527452552559164329625091, −6.34307560320525915669872499015, −5.38677261224167027607381449148, −4.55083797475651139481127504856, −3.76670139338667314526271631750, −3.13352908739783390120440584303, −1.22116492764177397244299601523, 1.56373271096809404512043635080, 2.62287178525519226845103481225, 3.13323400073370687336146945923, 4.69553472719243925664247327114, 5.99283416292580919612046404787, 6.60335177267628717211780271924, 7.15686313696971675203270054552, 7.971801917069062244808183618562, 8.758029292418522682948167989881, 9.753118314788861650360692655855

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