Properties

Label 2-90e2-1.1-c1-0-51
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.85·7-s + 1.85·11-s + 0.854·13-s + 1.14·17-s + 2·19-s − 4.85·23-s + 3.70·29-s + 2.70·31-s − 5.85·37-s − 11.5·41-s + 0.854·43-s − 6.70·47-s + 1.14·49-s + 4.85·53-s + 1.14·59-s + 0.854·61-s − 7·67-s + 9·71-s + 2.70·73-s − 5.29·77-s + 11.7·79-s + 6.70·83-s − 12·89-s − 2.43·91-s − 10·97-s + 4.85·101-s + 0.145·103-s + ⋯
L(s)  = 1  − 1.07·7-s + 0.559·11-s + 0.236·13-s + 0.277·17-s + 0.458·19-s − 1.01·23-s + 0.688·29-s + 0.486·31-s − 0.962·37-s − 1.80·41-s + 0.130·43-s − 0.978·47-s + 0.163·49-s + 0.666·53-s + 0.149·59-s + 0.109·61-s − 0.855·67-s + 1.06·71-s + 0.316·73-s − 0.603·77-s + 1.31·79-s + 0.736·83-s − 1.27·89-s − 0.255·91-s − 1.01·97-s + 0.483·101-s + 0.0143·103-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 \( 1 + 2.85T + 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 - 0.854T + 13T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 4.85T + 23T^{2} \)
29 \( 1 - 3.70T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 + 5.85T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 0.854T + 43T^{2} \)
47 \( 1 + 6.70T + 47T^{2} \)
53 \( 1 - 4.85T + 53T^{2} \)
59 \( 1 - 1.14T + 59T^{2} \)
61 \( 1 - 0.854T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.39346184012550029902696551415, −6.63061148942583433269502199946, −6.29890984800307370889876874136, −5.43956053237466513284942347375, −4.66994673962814135437145276462, −3.65206030952553901224024069582, −3.31953752623595239579231421067, −2.25713055807056420647284684534, −1.21571641440977399214490882049, 0, 1.21571641440977399214490882049, 2.25713055807056420647284684534, 3.31953752623595239579231421067, 3.65206030952553901224024069582, 4.66994673962814135437145276462, 5.43956053237466513284942347375, 6.29890984800307370889876874136, 6.63061148942583433269502199946, 7.39346184012550029902696551415

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