Properties

Degree $2$
Conductor $1764$
Sign $-0.577 + 0.816i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s + 2i·5-s + 2.82i·8-s + 2.82·10-s − 7.07·13-s + 4.00·16-s − 2i·17-s − 4.00i·20-s + 25-s + 10.0i·26-s − 9.89i·29-s − 5.65i·32-s − 2.82·34-s + 12·37-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 0.894i·5-s + 1.00i·8-s + 0.894·10-s − 1.96·13-s + 1.00·16-s − 0.485i·17-s − 0.894i·20-s + 0.200·25-s + 1.96i·26-s − 1.83i·29-s − 1.00i·32-s − 0.485·34-s + 1.97·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.577 + 0.816i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9849348543\)
\(L(\frac12)\) \(\approx\) \(0.9849348543\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7.07T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 12T + 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12.7iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−9.497456836513005277281501049822, −8.259602376710048630558562084905, −7.53430224264892804657960042838, −6.73095876756720919594703733970, −5.56981111621362003000498615326, −4.73719201562354593869658620435, −3.85288770084505224453031272397, −2.66745170882070923274634382903, −2.29555633457514920523112979363, −0.42870500871592043268282414520, 1.09483860997592126830343195965, 2.74440393467039269855684966733, 4.12521246856687554544636401912, 4.87907470592395728979913472387, 5.38247457313729314879864001835, 6.44981964668487474632504286682, 7.25688583069308497763668962420, 7.934019858196403446279335392555, 8.684488491919867481334173091958, 9.449298616183838917414621344527

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