Properties

Label 2-42e2-3.2-c2-0-27
Degree $2$
Conductor $1764$
Sign $-0.577 - 0.816i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.55i·5-s − 9.27i·11-s − 4.24·13-s − 15.5i·17-s − 34.8·19-s + 17.7i·23-s − 66.3·25-s + 26.2i·29-s − 32.0·31-s + 55.3·37-s + 38.4i·41-s + 29.3·43-s − 68.2i·47-s + 1.08i·53-s − 88.6·55-s + ⋯
L(s)  = 1  − 1.91i·5-s − 0.843i·11-s − 0.326·13-s − 0.915i·17-s − 1.83·19-s + 0.772i·23-s − 2.65·25-s + 0.904i·29-s − 1.03·31-s + 1.49·37-s + 0.937i·41-s + 0.682·43-s − 1.45i·47-s + 0.0205i·53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4471655350\)
\(L(\frac12)\) \(\approx\) \(0.4471655350\)
\(L(2)\) 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 \( 1 + 9.55iT - 25T^{2} \)
11 \( 1 + 9.27iT - 121T^{2} \)
13 \( 1 + 4.24T + 169T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 + 34.8T + 361T^{2} \)
23 \( 1 - 17.7iT - 529T^{2} \)
29 \( 1 - 26.2iT - 841T^{2} \)
31 \( 1 + 32.0T + 961T^{2} \)
37 \( 1 - 55.3T + 1.36e3T^{2} \)
41 \( 1 - 38.4iT - 1.68e3T^{2} \)
43 \( 1 - 29.3T + 1.84e3T^{2} \)
47 \( 1 + 68.2iT - 2.20e3T^{2} \)
53 \( 1 - 1.08iT - 2.80e3T^{2} \)
59 \( 1 + 68.2iT - 3.48e3T^{2} \)
61 \( 1 - 47.6T + 3.72e3T^{2} \)
67 \( 1 + 8.65T + 4.48e3T^{2} \)
71 \( 1 - 95.7iT - 5.04e3T^{2} \)
73 \( 1 - 90.0T + 5.32e3T^{2} \)
79 \( 1 + 148.T + 6.24e3T^{2} \)
83 \( 1 - 88.8iT - 6.88e3T^{2} \)
89 \( 1 - 76.2iT - 7.92e3T^{2} \)
97 \( 1 - 12.7T + 9.40e3T^{2} \)
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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

−8.576397612200970721826347480737, −8.121851672062081422999710112073, −7.07661465550893096925849690848, −5.94348813631461434598721709763, −5.27634245068569499887123000388, −4.54157536877560183258290791832, −3.70703467345353664418581278477, −2.26337729490059930721476044829, −1.11363613402187220079407239634, −0.12140160820408946687575332703, 2.07535012365434887037994773936, 2.57446025297785277859077389674, 3.80865866240709194198520196079, 4.45263167181805015067251138923, 6.04277615644557054094177498811, 6.33482039325287121095720272998, 7.28776071364757120188160091971, 7.78002144953683897300984607748, 8.864064507149625755062680610226, 9.857328204334484458965987173512

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