Properties

Degree $2$
Conductor $1764$
Sign $0.755 - 0.654i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·13-s + 1.73i·19-s + 25-s − 1.73i·31-s + 37-s + 43-s − 67-s + 1.73i·73-s − 79-s − 1.73i·103-s + 109-s + ⋯
L(s)  = 1  + 1.73i·13-s + 1.73i·19-s + 25-s − 1.73i·31-s + 37-s + 43-s − 67-s + 1.73i·73-s − 79-s − 1.73i·103-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Motivic weight: \(0\)
Character: $\chi_{1764} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118390064\)
\(L(\frac12)\) \(\approx\) \(1.118390064\)
\(L(1)\) 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 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.546241743492637231301743564817, −8.865857949801841562228746163395, −7.990575680104474115606634308158, −7.22425857804538120847758920325, −6.34061604149370236202950670022, −5.67759528043369136618398569354, −4.43807658020438802710731254167, −3.91280339917285512320307417121, −2.57510274776060613681995637119, −1.50487015914138607533452316670, 0.942561592259345294468296927486, 2.63217218719953035275204927225, 3.26571334090267834472061865824, 4.61772661579888641664611156362, 5.24536026167022527772398936435, 6.19813868057511161059301623153, 7.08962852102827203835123985182, 7.79251351648400939042685222173, 8.694103321015126369281764338354, 9.257921809794012541669276835668

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