Properties

Label 2-90e2-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.93·7-s + 2.41·11-s − 2.90·13-s + 6.86·17-s − 4.17·19-s + 3.35·23-s − 5.19·29-s − 6.17·31-s − 7.84·37-s + 5.87·41-s − 4.93·43-s − 11.9·47-s + 17.3·49-s + 8.54·53-s + 1.05·59-s + 9.17·61-s − 4.05·67-s + 14.1·71-s − 2.02·73-s − 11.9·77-s + 6·79-s − 5.18·83-s − 3.09·89-s + 14.3·91-s − 0.882·97-s − 5.87·101-s − 9.87·103-s + ⋯
L(s)  = 1  − 1.86·7-s + 0.727·11-s − 0.806·13-s + 1.66·17-s − 0.958·19-s + 0.700·23-s − 0.964·29-s − 1.10·31-s − 1.28·37-s + 0.917·41-s − 0.752·43-s − 1.73·47-s + 2.47·49-s + 1.17·53-s + 0.136·59-s + 1.17·61-s − 0.495·67-s + 1.68·71-s − 0.237·73-s − 1.35·77-s + 0.675·79-s − 0.569·83-s − 0.327·89-s + 1.50·91-s − 0.0896·97-s − 0.584·101-s − 0.972·103-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: \(0\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.108496299\)
\(L(\frac12)\) \(\approx\) \(1.108496299\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.93T + 7T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 + 2.90T + 13T^{2} \)
17 \( 1 - 6.86T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 + 5.19T + 29T^{2} \)
31 \( 1 + 6.17T + 31T^{2} \)
37 \( 1 + 7.84T + 37T^{2} \)
41 \( 1 - 5.87T + 41T^{2} \)
43 \( 1 + 4.93T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 8.54T + 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 - 9.17T + 61T^{2} \)
67 \( 1 + 4.05T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 2.02T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 5.18T + 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + 0.882T + 97T^{2} \)
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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

−7.70407376659421797518512387808, −6.89409637826444135633374071674, −6.68100001013953501218825842131, −5.70592725677384539952663653640, −5.25223310722367787596895784468, −4.00714321954654544741715418423, −3.51267830097585264236460285626, −2.85322905592995418293120666746, −1.79029508498385797437941119610, −0.50493238732193522933945870652, 0.50493238732193522933945870652, 1.79029508498385797437941119610, 2.85322905592995418293120666746, 3.51267830097585264236460285626, 4.00714321954654544741715418423, 5.25223310722367787596895784468, 5.70592725677384539952663653640, 6.68100001013953501218825842131, 6.89409637826444135633374071674, 7.70407376659421797518512387808

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