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Properties

Label 2-42e2-9.7-c1-0-14
Degree $2$
Conductor $1764$
Sign $0.552 - 0.833i$
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  + (1.69 + 0.353i)3-s + (−0.469 − 0.812i)5-s + (2.75 + 1.19i)9-s + (−1.31 + 2.28i)11-s + (2.71 + 4.69i)13-s + (−0.508 − 1.54i)15-s − 3.85·17-s + 1.09·19-s + (3.16 + 5.48i)23-s + (2.05 − 3.56i)25-s + (4.24 + 3.00i)27-s + (−1.94 + 3.36i)29-s + (−2.33 − 4.04i)31-s + (−3.04 + 3.40i)33-s + 2.30·37-s + ⋯
L(s)  = 1  + (0.979 + 0.203i)3-s + (−0.209 − 0.363i)5-s + (0.916 + 0.399i)9-s + (−0.397 + 0.688i)11-s + (0.751 + 1.30i)13-s + (−0.131 − 0.398i)15-s − 0.934·17-s + 0.251·19-s + (0.660 + 1.14i)23-s + (0.411 − 0.713i)25-s + (0.816 + 0.577i)27-s + (−0.360 + 0.624i)29-s + (−0.419 − 0.725i)31-s + (−0.529 + 0.592i)33-s + 0.379·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.323182724\)
\(L(\frac12)\) \(\approx\) \(2.323182724\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.353i)T \)
7 \( 1 \)
good5 \( 1 + (0.469 + 0.812i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.31 - 2.28i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.71 - 4.69i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.85T + 17T^{2} \)
19 \( 1 - 1.09T + 19T^{2} \)
23 \( 1 + (-3.16 - 5.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.94 - 3.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.33 + 4.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + (-4.12 - 7.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.14 - 3.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.32 - 2.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + (3.02 + 5.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.71 + 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.64 - 6.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 5.74T + 73T^{2} \)
79 \( 1 + (5.51 - 9.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.24 + 2.15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (1.75 - 3.03i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.427779060536841798490732325669, −8.669649313884591369802831103757, −7.980701999388014397031085282126, −7.14708933107019091876372777572, −6.43536191947359168220630231036, −5.05671385885169713745553396068, −4.37649119714808898729819788378, −3.59100988717328028539353230476, −2.42465255342961919209389013634, −1.47259563181071873567554065750, 0.813487008547812658610409056102, 2.33843160984833997121287921286, 3.15728098356505063822134591280, 3.86061546320768506255029736645, 5.06106711036829587451107764582, 6.05885476538678679749706450730, 6.97823145773251972855698808731, 7.64679560812526509623581273071, 8.558838601422115407081176858887, 8.818785013937507570989210043563

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