Properties

Label 2-90e2-1.1-c1-0-56
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  + 2.73·7-s − 1.73·11-s − 5.46·13-s − 4.73·17-s − 4.46·19-s + 3.46·23-s + 7.73·29-s + 5.92·31-s + 6.19·37-s + 11.1·41-s − 3.26·43-s − 1.26·47-s + 0.464·49-s − 7.26·53-s + 7.73·59-s − 4·61-s − 6.39·67-s + 11.1·71-s + 0.196·73-s − 4.73·77-s − 14.3·79-s − 15.1·83-s − 5.19·89-s − 14.9·91-s − 0.732·97-s − 6.12·101-s − 18.3·103-s + ⋯
L(s)  = 1  + 1.03·7-s − 0.522·11-s − 1.51·13-s − 1.14·17-s − 1.02·19-s + 0.722·23-s + 1.43·29-s + 1.06·31-s + 1.01·37-s + 1.74·41-s − 0.498·43-s − 0.184·47-s + 0.0663·49-s − 0.998·53-s + 1.00·59-s − 0.512·61-s − 0.780·67-s + 1.32·71-s + 0.0229·73-s − 0.539·77-s − 1.61·79-s − 1.66·83-s − 0.550·89-s − 1.56·91-s − 0.0743·97-s − 0.609·101-s − 1.81·103-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 \( 1 - 2.73T + 7T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 - 6.19T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 3.26T + 43T^{2} \)
47 \( 1 + 1.26T + 47T^{2} \)
53 \( 1 + 7.26T + 53T^{2} \)
59 \( 1 - 7.73T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 6.39T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 0.196T + 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + 0.732T + 97T^{2} \)
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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

−7.52823112670035452477514657938, −6.83311705257973475996541678650, −6.17407753021035478280464558705, −5.17677881684802930378837505420, −4.61434920652647565414799601207, −4.26832180120372390562998740704, −2.67166675170214527963382748820, −2.49163737578087806360969116730, −1.28578020329537963669087552742, 0, 1.28578020329537963669087552742, 2.49163737578087806360969116730, 2.67166675170214527963382748820, 4.26832180120372390562998740704, 4.61434920652647565414799601207, 5.17677881684802930378837505420, 6.17407753021035478280464558705, 6.83311705257973475996541678650, 7.52823112670035452477514657938

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