Properties

Label 2-84e2-1.1-c1-0-38
Degree $2$
Conductor $7056$
Sign $1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·5-s − 2·11-s + 5.65·13-s + 2.82·17-s − 5.65·19-s − 6·23-s + 3.00·25-s − 4·29-s + 5.65·31-s − 2·37-s − 2.82·41-s + 4·43-s + 11.3·47-s + 12·53-s − 5.65·55-s + 11.3·59-s + 5.65·61-s + 16.0·65-s + 12·67-s + 6·71-s − 8·79-s − 11.3·83-s + 8.00·85-s + 8.48·89-s − 16.0·95-s − 11.3·97-s + 8.48·101-s + ⋯
L(s)  = 1  + 1.26·5-s − 0.603·11-s + 1.56·13-s + 0.685·17-s − 1.29·19-s − 1.25·23-s + 0.600·25-s − 0.742·29-s + 1.01·31-s − 0.328·37-s − 0.441·41-s + 0.609·43-s + 1.65·47-s + 1.64·53-s − 0.762·55-s + 1.47·59-s + 0.724·61-s + 1.98·65-s + 1.46·67-s + 0.712·71-s − 0.900·79-s − 1.24·83-s + 0.867·85-s + 0.899·89-s − 1.64·95-s − 1.14·97-s + 0.844·101-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.727699242\)
\(L(\frac12)\) \(\approx\) \(2.727699242\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 2.82T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 8.48T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.191245684407558222333873557556, −7.12536027614820631411796659282, −6.39114218894079728637947381454, −5.73614789538599436254478027254, −5.48821545586914128176805085498, −4.24343111771904064186306611382, −3.65653234180789986182151587459, −2.47589361776507879777210754850, −1.93617327452298501371290145868, −0.853915897345836822854217443224, 0.853915897345836822854217443224, 1.93617327452298501371290145868, 2.47589361776507879777210754850, 3.65653234180789986182151587459, 4.24343111771904064186306611382, 5.48821545586914128176805085498, 5.73614789538599436254478027254, 6.39114218894079728637947381454, 7.12536027614820631411796659282, 8.191245684407558222333873557556

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