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Properties

Label 2-42e2-63.25-c1-0-9
Degree $2$
Conductor $1764$
Sign $0.702 - 0.711i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.619 − 1.61i)3-s + (−0.119 + 0.207i)5-s + (−2.23 − 2.00i)9-s + (2.56 + 4.43i)11-s + (2.44 + 4.23i)13-s + (0.260 + 0.321i)15-s + (−1.85 + 3.20i)17-s + (−1.83 − 3.16i)19-s + (−3.71 + 6.42i)23-s + (2.47 + 4.28i)25-s + (−4.62 + 2.36i)27-s + (−1.73 + 3.00i)29-s − 0.717·31-s + (8.76 − 1.39i)33-s + (−2.30 − 3.98i)37-s + ⋯
L(s)  = 1  + (0.357 − 0.933i)3-s + (−0.0534 + 0.0926i)5-s + (−0.744 − 0.668i)9-s + (0.772 + 1.33i)11-s + (0.677 + 1.17i)13-s + (0.0673 + 0.0830i)15-s + (−0.449 + 0.777i)17-s + (−0.419 − 0.727i)19-s + (−0.773 + 1.34i)23-s + (0.494 + 0.856i)25-s + (−0.890 + 0.455i)27-s + (−0.321 + 0.557i)29-s − 0.128·31-s + (1.52 − 0.242i)33-s + (−0.378 − 0.655i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.611754342\)
\(L(\frac12)\) \(\approx\) \(1.611754342\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 + 1.61i)T \)
7 \( 1 \)
good5 \( 1 + (0.119 - 0.207i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.56 - 4.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.85 - 3.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 + 3.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.71 - 6.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.73 - 3.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.717T + 31T^{2} \)
37 \( 1 + (2.30 + 3.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.24 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.33T + 47T^{2} \)
53 \( 1 + (-0.471 + 0.816i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.57T + 59T^{2} \)
61 \( 1 - 5.50T + 61T^{2} \)
67 \( 1 + 0.660T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + (-1.83 + 3.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.22T + 79T^{2} \)
83 \( 1 + (4.85 - 8.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.74 - 6.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.159876203759779596264218738812, −8.731792662955092209816804416003, −7.69184405392301416807712854832, −6.89020051527465148280874316034, −6.58614917263065517262714119539, −5.46898163833970433810389969026, −4.23968076682299230552723010106, −3.52342424899645714582401170086, −2.05774360937792651331486295428, −1.52123844052684382105006510488, 0.57655025910774539293761534121, 2.38915201023572186298100851008, 3.41109568997960740313877013525, 4.02536913713606783732089852095, 5.07129276345924076061857046402, 5.91423870640694321721846802449, 6.61759054342312237728576914560, 8.072132320512130907561014704871, 8.458311677896346351008796976045, 9.023208323479591392941454736046

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