Properties

Label 2-42e2-7.6-c2-0-22
Degree $2$
Conductor $1764$
Sign $0.156 + 0.987i$
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  − 0.929i·5-s − 9.68·11-s + 15.9i·13-s − 10.5i·17-s − 7.22i·19-s + 11.3·23-s + 24.1·25-s − 46.3·29-s − 0.483i·31-s + 2.48·37-s − 55.8i·41-s + 60.6·43-s + 36.5i·47-s + 28.5·53-s + 9.00i·55-s + ⋯
L(s)  = 1  − 0.185i·5-s − 0.880·11-s + 1.22i·13-s − 0.620i·17-s − 0.380i·19-s + 0.491·23-s + 0.965·25-s − 1.59·29-s − 0.0155i·31-s + 0.0670·37-s − 1.36i·41-s + 1.41·43-s + 0.778i·47-s + 0.538·53-s + 0.163i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.336127784\)
\(L(\frac12)\) \(\approx\) \(1.336127784\)
\(L(2)\) 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 \( 1 + 0.929iT - 25T^{2} \)
11 \( 1 + 9.68T + 121T^{2} \)
13 \( 1 - 15.9iT - 169T^{2} \)
17 \( 1 + 10.5iT - 289T^{2} \)
19 \( 1 + 7.22iT - 361T^{2} \)
23 \( 1 - 11.3T + 529T^{2} \)
29 \( 1 + 46.3T + 841T^{2} \)
31 \( 1 + 0.483iT - 961T^{2} \)
37 \( 1 - 2.48T + 1.36e3T^{2} \)
41 \( 1 + 55.8iT - 1.68e3T^{2} \)
43 \( 1 - 60.6T + 1.84e3T^{2} \)
47 \( 1 - 36.5iT - 2.20e3T^{2} \)
53 \( 1 - 28.5T + 2.80e3T^{2} \)
59 \( 1 + 94.0iT - 3.48e3T^{2} \)
61 \( 1 + 110. iT - 3.72e3T^{2} \)
67 \( 1 - 82.0T + 4.48e3T^{2} \)
71 \( 1 + 127.T + 5.04e3T^{2} \)
73 \( 1 + 46.2iT - 5.32e3T^{2} \)
79 \( 1 - 18.7T + 6.24e3T^{2} \)
83 \( 1 - 59.6iT - 6.88e3T^{2} \)
89 \( 1 + 71.1iT - 7.92e3T^{2} \)
97 \( 1 - 102. iT - 9.40e3T^{2} \)
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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

−9.235008391869103656432627458250, −8.126402057007061495602555256448, −7.30359450351799445828836770585, −6.68517764106042495566330851709, −5.56224994508310896569421587180, −4.88447047680753351896067733690, −3.96887282619269989060046351399, −2.82954233810510214638255459755, −1.85371833843514582864631016371, −0.39711982019141048411410828179, 1.02894165830333836472010570570, 2.45198050233381844880525203811, 3.25632815236009787288309875633, 4.30897859277836407013585214565, 5.42222754504183106919850466086, 5.86433142084335624790037960303, 7.06363453620470817460154982219, 7.71142752037606017569936677713, 8.432090796241675810833749519832, 9.252740879820639605921599868593

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