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Properties

Label 2-38e2-19.10-c0-0-1
Degree $2$
Conductor $1444$
Sign $-0.322 - 0.946i$
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  + (0.909 + 1.08i)3-s + (−0.939 − 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.483 − 1.32i)15-s + (0.173 + 0.984i)17-s + (−1.39 + 0.245i)21-s + (−1.39 − 0.245i)29-s + (−0.483 + 1.32i)33-s + (0.766 − 0.642i)35-s + 1.41i·37-s + (−0.909 − 1.08i)41-s + (0.939 + 0.342i)43-s + (0.499 − 0.866i)45-s + ⋯
L(s)  = 1  + (0.909 + 1.08i)3-s + (−0.939 − 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.483 − 1.32i)15-s + (0.173 + 0.984i)17-s + (−1.39 + 0.245i)21-s + (−1.39 − 0.245i)29-s + (−0.483 + 1.32i)33-s + (0.766 − 0.642i)35-s + 1.41i·37-s + (−0.909 − 1.08i)41-s + (0.939 + 0.342i)43-s + (0.499 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.322 - 0.946i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.322 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131344653\)
\(L(\frac12)\) \(\approx\) \(1.131344653\)
\(L(1)\) 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 \( 1 + (-0.909 - 1.08i)T + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
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 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (1.39 + 0.245i)T + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + (0.909 + 1.08i)T + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-1.39 + 0.245i)T + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.483 + 1.32i)T + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−9.678156358745094472482921734472, −9.258980037866226654785812085117, −8.449053366026061882758515260262, −7.904765740992183196592898068643, −6.79304063528920424867343829425, −5.71283102316978855505650553900, −4.62408190844240495493910898151, −3.94333416954828710046895612926, −3.26570676132359359108466865379, −2.05429710546338401578347421757, 0.855263366079335504856861325321, 2.35637963161640333326001896865, 3.48313591264527052823470873489, 3.85868360407537993444868672243, 5.43122786475661636662479449665, 6.66882503712794718796894062190, 7.20011229344673062748082838703, 7.70007214699566940580252208417, 8.530716301478695198537928322656, 9.252778054365581168605462585066

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