Properties

Label 2-38e2-19.15-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.672 - 0.740i$
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  + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)17-s + (1.53 + 1.28i)23-s + (0.173 + 0.984i)35-s + (−0.766 + 0.642i)43-s + (0.5 − 0.866i)45-s + (0.939 − 0.342i)47-s + (−0.939 − 0.342i)55-s + (−0.766 − 0.642i)61-s + (−0.173 + 0.984i)63-s + (−0.173 − 0.984i)73-s + 0.999·77-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)7-s + (−0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (0.939 + 0.342i)17-s + (1.53 + 1.28i)23-s + (0.173 + 0.984i)35-s + (−0.766 + 0.642i)43-s + (0.5 − 0.866i)45-s + (0.939 − 0.342i)47-s + (−0.939 − 0.342i)55-s + (−0.766 − 0.642i)61-s + (−0.173 + 0.984i)63-s + (−0.173 − 0.984i)73-s + 0.999·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.672 - 0.740i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9661500862\)
\(L(\frac12)\) \(\approx\) \(0.9661500862\)
\(L(1)\) 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 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.766 - 0.642i)T^{2} \)
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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

−9.886940712231416103057073376773, −9.015351979790773241431444824681, −7.985316922466200420419025459704, −7.45966791389632421992738826988, −6.84605813483333681213953026707, −5.64054992015472729597704667910, −4.74564587462473195367746254399, −3.75130509486854263393211824391, −2.99790497160659598509919664440, −1.46007341096216204260244962391, 0.903241290934177371385519360749, 2.63241949588713765969114059666, 3.50713896686204169279568213297, 4.65806610128235004011945769301, 5.44707631180684654849215423146, 6.19314474527644477021500015214, 7.29665723921618973831742696989, 8.396830109110692318965843025206, 8.612875827655572386000674657600, 9.278978413018265241485753811707

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