Properties

Label 2-42e2-3.2-c2-0-23
Degree $2$
Conductor $1764$
Sign $-0.577 + 0.816i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
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  − 9.55i·5-s + 9.27i·11-s + 4.24·13-s − 15.5i·17-s + 34.8·19-s − 17.7i·23-s − 66.3·25-s − 26.2i·29-s + 32.0·31-s + 55.3·37-s + 38.4i·41-s + 29.3·43-s − 68.2i·47-s − 1.08i·53-s + 88.6·55-s + ⋯
L(s)  = 1  − 1.91i·5-s + 0.843i·11-s + 0.326·13-s − 0.915i·17-s + 1.83·19-s − 0.772i·23-s − 2.65·25-s − 0.904i·29-s + 1.03·31-s + 1.49·37-s + 0.937i·41-s + 0.682·43-s − 1.45i·47-s − 0.0205i·53-s + 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.910814700\)
\(L(\frac12)\) \(\approx\) \(1.910814700\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 9.55iT - 25T^{2} \)
11 \( 1 - 9.27iT - 121T^{2} \)
13 \( 1 - 4.24T + 169T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 - 34.8T + 361T^{2} \)
23 \( 1 + 17.7iT - 529T^{2} \)
29 \( 1 + 26.2iT - 841T^{2} \)
31 \( 1 - 32.0T + 961T^{2} \)
37 \( 1 - 55.3T + 1.36e3T^{2} \)
41 \( 1 - 38.4iT - 1.68e3T^{2} \)
43 \( 1 - 29.3T + 1.84e3T^{2} \)
47 \( 1 + 68.2iT - 2.20e3T^{2} \)
53 \( 1 + 1.08iT - 2.80e3T^{2} \)
59 \( 1 + 68.2iT - 3.48e3T^{2} \)
61 \( 1 + 47.6T + 3.72e3T^{2} \)
67 \( 1 + 8.65T + 4.48e3T^{2} \)
71 \( 1 + 95.7iT - 5.04e3T^{2} \)
73 \( 1 + 90.0T + 5.32e3T^{2} \)
79 \( 1 + 148.T + 6.24e3T^{2} \)
83 \( 1 - 88.8iT - 6.88e3T^{2} \)
89 \( 1 - 76.2iT - 7.92e3T^{2} \)
97 \( 1 + 12.7T + 9.40e3T^{2} \)
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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

−8.925866437590504733372524295439, −8.042681467682698428367384345588, −7.53044108164906006157258272142, −6.31975384679418301992242333378, −5.35478497045439804726839410458, −4.76733081547474464269974726292, −4.09489927392412180157615333567, −2.67996748793577988485862282488, −1.37791767062607346548234643920, −0.56174532089995450249990047182, 1.30655400062952247127484198251, 2.81213703765693816030004712712, 3.21757506687676534129070232299, 4.18919288883774549342337071300, 5.80648651075254365263452510990, 6.00547967871380880639923961157, 7.16750694840821125077993524909, 7.54675790040656112661314102796, 8.527759234309867066272486471951, 9.576201072254436427963252598530

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