Properties

Label 2-42e2-21.20-c1-0-6
Degree $2$
Conductor $1764$
Sign $0.860 + 0.508i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.84·5-s − 2i·11-s + 4.46i·13-s + 2.29·17-s − 1.53i·19-s − 8.82i·23-s − 1.58·25-s + 1.17i·29-s + 5.86i·31-s + 8.24·37-s + 11.8·41-s + 1.17·43-s − 8.02·47-s − 3.75i·53-s + 3.69i·55-s + ⋯
L(s)  = 1  − 0.826·5-s − 0.603i·11-s + 1.23i·13-s + 0.556·17-s − 0.351i·19-s − 1.84i·23-s − 0.317·25-s + 0.217i·29-s + 1.05i·31-s + 1.35·37-s + 1.85·41-s + 0.178·43-s − 1.17·47-s − 0.516i·53-s + 0.498i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.344379571\)
\(L(\frac12)\) \(\approx\) \(1.344379571\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.84T + 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 4.46iT - 13T^{2} \)
17 \( 1 - 2.29T + 17T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
23 \( 1 + 8.82iT - 23T^{2} \)
29 \( 1 - 1.17iT - 29T^{2} \)
31 \( 1 - 5.86iT - 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 1.17T + 43T^{2} \)
47 \( 1 + 8.02T + 47T^{2} \)
53 \( 1 + 3.75iT - 53T^{2} \)
59 \( 1 - 9.81T + 59T^{2} \)
61 \( 1 + 12.3iT - 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + 2.74iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 - 2.74iT - 97T^{2} \)
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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

−9.160549175964570283228003009489, −8.375419836597499739022617400301, −7.77616247254958757086689270662, −6.78349847947643574901816602334, −6.20674909417279214676383829796, −4.97194947451148259310408377545, −4.23477527593013707140037839779, −3.37279743824186146531828830380, −2.23331809833142850814003886467, −0.67520443543456329616929782731, 0.949516211440669714084407610682, 2.46605597469200169521236291768, 3.56488869578384213870549419969, 4.23383352685307655859618784176, 5.40540931249902720108902584277, 5.98934434509120405241661162118, 7.34124198902901756479013746340, 7.70887308301174859496874300826, 8.337149629047287482227228425420, 9.651460758930827215050605941255

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