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Properties

Label 2-84e2-28.27-c1-0-65
Degree $2$
Conductor $7056$
Sign $-0.156 + 0.987i$
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  − 1.84i·5-s + 5.22i·11-s − 4.46i·13-s + 2.93i·17-s − 5.65·19-s + 2.16i·23-s + 1.58·25-s + 5.41·29-s − 9.65·31-s + 1.41·37-s − 4.01i·41-s − 3.06i·43-s + 1.65·47-s + 9.65·53-s + 9.65·55-s + ⋯
L(s)  = 1  − 0.826i·5-s + 1.57i·11-s − 1.23i·13-s + 0.710i·17-s − 1.29·19-s + 0.451i·23-s + 0.317·25-s + 1.00·29-s − 1.73·31-s + 0.232·37-s − 0.626i·41-s − 0.466i·43-s + 0.241·47-s + 1.32·53-s + 1.30·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.307420422\)
\(L(\frac12)\) \(\approx\) \(1.307420422\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.84iT - 5T^{2} \)
11 \( 1 - 5.22iT - 11T^{2} \)
13 \( 1 + 4.46iT - 13T^{2} \)
17 \( 1 - 2.93iT - 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 2.16iT - 23T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 + 4.01iT - 41T^{2} \)
43 \( 1 + 3.06iT - 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 - 9.65T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 - 1.39iT - 61T^{2} \)
67 \( 1 + 13.5iT - 67T^{2} \)
71 \( 1 + 6.49iT - 71T^{2} \)
73 \( 1 + 4.90iT - 73T^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 + 9.05iT - 89T^{2} \)
97 \( 1 + 9.23iT - 97T^{2} \)
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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

−7.66428548275588916318236585386, −7.22635981182506804435203348991, −6.29953232252976474082108036150, −5.54880165090846121889997628403, −4.87454398030004273613825272847, −4.26947655209836950322101498611, −3.44796739983044665522482862290, −2.28921698772796201115325550211, −1.60042284555499305517886336265, −0.35672604590437313729446746277, 0.974574006829780575671548514610, 2.27828365493926820180620134315, 2.87857133310939889219218833468, 3.78947525640401213390185398564, 4.42084679079463194038192637219, 5.45476068134940591371855808267, 6.12162908478566941292081286982, 6.83989640913272651524038286181, 7.14321511560635473405446889651, 8.401199883356228869916988048105

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