Properties

Label 2-90e2-1.1-c1-0-43
Degree $2$
Conductor $8100$
Sign $-1$
Analytic cond. $64.6788$
Root an. cond. $8.04231$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.26·7-s − 2.54·11-s − 5.02·13-s + 5.72·17-s − 1.86·19-s + 5.39·23-s − 3·29-s + 9.38·31-s − 2.59·37-s + 3.96·41-s + 10.2·43-s + 1.07·47-s − 1.86·49-s + 10.7·53-s − 1.58·59-s + 0.869·61-s − 4.85·67-s − 1.86·71-s + 3.87·73-s + 5.76·77-s − 13.2·79-s − 14.7·83-s − 13.4·89-s + 11.3·91-s + 17.6·97-s − 5.32·101-s − 6.01·103-s + ⋯
L(s)  = 1  − 0.856·7-s − 0.768·11-s − 1.39·13-s + 1.38·17-s − 0.429·19-s + 1.12·23-s − 0.557·29-s + 1.68·31-s − 0.426·37-s + 0.619·41-s + 1.55·43-s + 0.156·47-s − 0.267·49-s + 1.47·53-s − 0.206·59-s + 0.111·61-s − 0.593·67-s − 0.221·71-s + 0.453·73-s + 0.657·77-s − 1.49·79-s − 1.61·83-s − 1.42·89-s + 1.19·91-s + 1.79·97-s − 0.529·101-s − 0.592·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(64.6788\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 + 2.54T + 11T^{2} \)
13 \( 1 + 5.02T + 13T^{2} \)
17 \( 1 - 5.72T + 17T^{2} \)
19 \( 1 + 1.86T + 19T^{2} \)
23 \( 1 - 5.39T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 9.38T + 31T^{2} \)
37 \( 1 + 2.59T + 37T^{2} \)
41 \( 1 - 3.96T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 1.07T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 1.58T + 59T^{2} \)
61 \( 1 - 0.869T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + 1.86T + 71T^{2} \)
73 \( 1 - 3.87T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−7.40446627056329135040579039071, −6.95462786897500960040365129758, −6.01314382036260459640905693946, −5.42974233086345344737944844991, −4.72808012585118174423156865571, −3.88112979007498937617870549647, −2.82217552805318393691723962545, −2.60539910398028075536643206434, −1.12656824739779404097443394110, 0, 1.12656824739779404097443394110, 2.60539910398028075536643206434, 2.82217552805318393691723962545, 3.88112979007498937617870549647, 4.72808012585118174423156865571, 5.42974233086345344737944844991, 6.01314382036260459640905693946, 6.95462786897500960040365129758, 7.40446627056329135040579039071

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