Properties

Label 2-84e2-21.20-c1-0-76
Degree $2$
Conductor $7056$
Sign $-0.860 + 0.508i$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  + 2.93·5-s − 4.82i·11-s − 2.93i·13-s − 7.07·17-s + 5.86i·19-s + 2i·23-s + 3.58·25-s − 0.828i·29-s − 5.86i·31-s − 5.41·37-s − 1.21·41-s − 4.48·43-s − 5.86·47-s − 7.07i·53-s − 14.1i·55-s + ⋯
L(s)  = 1  + 1.31·5-s − 1.45i·11-s − 0.812i·13-s − 1.71·17-s + 1.34i·19-s + 0.417i·23-s + 0.717·25-s − 0.153i·29-s − 1.05i·31-s − 0.890·37-s − 0.189·41-s − 0.683·43-s − 0.854·47-s − 0.971i·53-s − 1.90i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.860 + 0.508i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.062968064\)
\(L(\frac12)\) \(\approx\) \(1.062968064\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.93T + 5T^{2} \)
11 \( 1 + 4.82iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 - 5.86iT - 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 0.828iT - 29T^{2} \)
31 \( 1 + 5.86iT - 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + 5.86T + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 5.86T + 59T^{2} \)
61 \( 1 + 1.21iT - 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 + 1.65T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 7.07iT - 97T^{2} \)
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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

−7.82634831998153547132515519601, −6.65917136763507031112512928275, −6.22501990936437933274499245487, −5.61453436421438465226649151862, −5.05769339439136114007281728160, −3.91352171300998701677698260003, −3.19069248639252601088189347410, −2.25977559065271480881644531148, −1.52579825895698779457666885764, −0.22121468390301595157909422831, 1.52751249273918298430174049297, 2.12488819413898098326702039720, 2.77269385001125963418677357324, 4.11361385221057290597896058277, 4.80327709205191955623009993370, 5.21825490896335843960177299660, 6.44363736333176773157916560369, 6.69712527399416536089036999644, 7.23520391685886007434347205969, 8.414701539781055863048404028743

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