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Properties

Label 2-38e2-19.12-c0-0-0
Degree $2$
Conductor $1444$
Sign $-0.305 - 0.952i$
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  + (0.5 + 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)17-s + (−1 + 1.73i)23-s + (−0.5 − 0.866i)35-s + (0.5 + 0.866i)43-s − 0.999·45-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)55-s + (0.5 − 0.866i)61-s + (0.5 − 0.866i)63-s + (0.5 + 0.866i)73-s + 77-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)17-s + (−1 + 1.73i)23-s + (−0.5 − 0.866i)35-s + (0.5 + 0.866i)43-s − 0.999·45-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)55-s + (0.5 − 0.866i)61-s + (0.5 − 0.866i)63-s + (0.5 + 0.866i)73-s + 77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7985746579\)
\(L(\frac12)\) \(\approx\) \(0.7985746579\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.07357511585902444549712042577, −9.378425387187765761140308637221, −8.159199594605645473307960865990, −7.64468821262505167670420497366, −6.61236037191190504224677213673, −5.86864876691273257795639999140, −5.23553128078275267156677108279, −3.76164804416741179238826528102, −2.91333630617745562481150830755, −2.04427352570763004950562669902, 0.61502814383840666277453031149, 2.40290751585530755382738396254, 3.28912761803733197402161722223, 4.47181732868480573484273247992, 5.45530975981446613030679895909, 6.08622087369974846490588521189, 6.94543881930385196867850818075, 8.013120238370716009915483029140, 8.829971732467341798668091817329, 9.450894102198111153581618043208

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