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Properties

Label 2-42e2-63.25-c1-0-33
Degree $2$
Conductor $1764$
Sign $-0.0477 + 0.998i$
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  + (1.5 − 0.866i)3-s + (1 − 1.73i)5-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + (1.5 + 2.59i)13-s − 3.46i·15-s + (3.5 − 6.06i)17-s + (2.5 + 4.33i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s − 5.19i·27-s + (0.5 − 0.866i)29-s + 3·31-s + (−6 − 3.46i)33-s + (−5.5 − 9.52i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.447 − 0.774i)5-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + (0.416 + 0.720i)13-s − 0.894i·15-s + (0.848 − 1.47i)17-s + (0.573 + 0.993i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s − 0.999i·27-s + (0.0928 − 0.160i)29-s + 0.538·31-s + (−1.04 − 0.603i)33-s + (−0.904 − 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0477 + 0.998i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.0477 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.538633943\)
\(L(\frac12)\) \(\approx\) \(2.538633943\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.120807777214719620873676188780, −8.301774004122221458064652006628, −7.69007739914466243808242090016, −6.84483298123254027906115473969, −5.75089422604068289041166831504, −5.18742021201644664951498607798, −3.82869510926641463380100602001, −3.09454632223976632405438671401, −1.89777830414191377740275437163, −0.882924356690714378506897726111, 1.68309261561478016638137928214, 2.73960157951384522743806997312, 3.37462675618557716701940182390, 4.52135394542479127503723948004, 5.31365696501857716759121564615, 6.42613742218254134846892595742, 7.16951288624935488429464463372, 8.166892114151927444655121758420, 8.496231494552671455275119486478, 9.816832904352433644236642448105

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