Properties

Label 2-90e2-1.1-c1-0-14
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  − 3.76·7-s + 3.54·11-s + 1.00·13-s − 1.56·17-s + 7.21·19-s − 6.18·23-s + 3·29-s − 5.78·31-s + 0.851·37-s + 3.11·41-s − 2.70·43-s − 9.74·47-s + 7.21·49-s + 11.2·53-s + 9.66·59-s − 8.21·61-s − 2.91·67-s − 7.21·71-s + 16.7·73-s − 13.3·77-s + 10.9·79-s − 10.1·83-s + 5.33·89-s − 3.78·91-s − 3.86·97-s − 4.75·101-s + 18.0·103-s + ⋯
L(s)  = 1  − 1.42·7-s + 1.06·11-s + 0.278·13-s − 0.378·17-s + 1.65·19-s − 1.28·23-s + 0.557·29-s − 1.03·31-s + 0.139·37-s + 0.487·41-s − 0.412·43-s − 1.42·47-s + 1.03·49-s + 1.54·53-s + 1.25·59-s − 1.05·61-s − 0.356·67-s − 0.855·71-s + 1.96·73-s − 1.52·77-s + 1.23·79-s − 1.11·83-s + 0.565·89-s − 0.396·91-s − 0.392·97-s − 0.473·101-s + 1.78·103-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

\(L(1)\) \(\approx\) \(1.599261079\)
\(L(\frac12)\) \(\approx\) \(1.599261079\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.76T + 7T^{2} \)
11 \( 1 - 3.54T + 11T^{2} \)
13 \( 1 - 1.00T + 13T^{2} \)
17 \( 1 + 1.56T + 17T^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 5.78T + 31T^{2} \)
37 \( 1 - 0.851T + 37T^{2} \)
41 \( 1 - 3.11T + 41T^{2} \)
43 \( 1 + 2.70T + 43T^{2} \)
47 \( 1 + 9.74T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 9.66T + 59T^{2} \)
61 \( 1 + 8.21T + 61T^{2} \)
67 \( 1 + 2.91T + 67T^{2} \)
71 \( 1 + 7.21T + 71T^{2} \)
73 \( 1 - 16.7T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 5.33T + 89T^{2} \)
97 \( 1 + 3.86T + 97T^{2} \)
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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

−7.73664629027283628174820704399, −7.01059009073341886407003413774, −6.45600363966897213530433130030, −5.90507147834810161585997002014, −5.11856558337338705537958336784, −4.01697012738137716315343844246, −3.57966878236303592534894235745, −2.82256577927961480646493927618, −1.72570968378819984693960096647, −0.62724341369265496928161540937, 0.62724341369265496928161540937, 1.72570968378819984693960096647, 2.82256577927961480646493927618, 3.57966878236303592534894235745, 4.01697012738137716315343844246, 5.11856558337338705537958336784, 5.90507147834810161585997002014, 6.45600363966897213530433130030, 7.01059009073341886407003413774, 7.73664629027283628174820704399

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