Properties

Degree $2$
Conductor $1764$
Sign $0.605 - 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)5-s + (1 + 1.73i)11-s − 4·13-s + (3 + 5.19i)17-s + (−4 + 6.92i)19-s + (−3 + 5.19i)23-s + (0.500 + 0.866i)25-s + 10·29-s + (−2 − 3.46i)31-s + (−3 + 5.19i)37-s + 6·41-s + 4·43-s + (4 − 6.92i)47-s + (1 + 1.73i)53-s + 3.99·55-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.301 + 0.522i)11-s − 1.10·13-s + (0.727 + 1.26i)17-s + (−0.917 + 1.58i)19-s + (−0.625 + 1.08i)23-s + (0.100 + 0.173i)25-s + 1.85·29-s + (−0.359 − 0.622i)31-s + (−0.493 + 0.854i)37-s + 0.937·41-s + 0.609·43-s + (0.583 − 1.01i)47-s + (0.137 + 0.237i)53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.596894815\)
\(L(\frac12)\) \(\approx\) \(1.596894815\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−9.657756893702986201759218320771, −8.491677782232552694231415706065, −8.027382386670021601496736507693, −7.05450527446159565153980669696, −6.04584554760532082087650235369, −5.43240275147190804650305990858, −4.45505166846529717027683983751, −3.66516411479278706742311765831, −2.20705405902293397210486293381, −1.33307648517861311081827693162, 0.62341083741715541956043776308, 2.52025647958306487466751152811, 2.79553370923834536353293614416, 4.29648927015540688758092502634, 5.04966969264627357157604158314, 6.10976962063568839156611912040, 6.82404610126947459825214127948, 7.39554198926179615503978294676, 8.509782118843760520007398093869, 9.193713381170703994465212294195

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