Properties

Label 2-51e2-1.1-c1-0-64
Degree $2$
Conductor $2601$
Sign $1$
Analytic cond. $20.7690$
Root an. cond. $4.55731$
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  + 2.30·2-s + 3.30·4-s + 2.30·5-s + 0.302·7-s + 3.00·8-s + 5.30·10-s + 3·11-s − 3.30·13-s + 0.697·14-s + 0.302·16-s + 5.90·19-s + 7.60·20-s + 6.90·22-s + 2.30·23-s + 0.302·25-s − 7.60·26-s + 1.00·28-s + 9.90·29-s − 3.60·31-s − 5.30·32-s + 0.697·35-s − 0.605·37-s + 13.6·38-s + 6.90·40-s + 6·41-s − 2.39·43-s + 9.90·44-s + ⋯
L(s)  = 1  + 1.62·2-s + 1.65·4-s + 1.02·5-s + 0.114·7-s + 1.06·8-s + 1.67·10-s + 0.904·11-s − 0.916·13-s + 0.186·14-s + 0.0756·16-s + 1.35·19-s + 1.70·20-s + 1.47·22-s + 0.480·23-s + 0.0605·25-s − 1.49·26-s + 0.188·28-s + 1.83·29-s − 0.647·31-s − 0.937·32-s + 0.117·35-s − 0.0995·37-s + 2.20·38-s + 1.09·40-s + 0.937·41-s − 0.365·43-s + 1.49·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(20.7690\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2601,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.939202100\)
\(L(\frac12)\) \(\approx\) \(5.939202100\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 - 2.30T + 2T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 - 0.302T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
19 \( 1 - 5.90T + 19T^{2} \)
23 \( 1 - 2.30T + 23T^{2} \)
29 \( 1 - 9.90T + 29T^{2} \)
31 \( 1 + 3.60T + 31T^{2} \)
37 \( 1 + 0.605T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 2.09T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 8.81T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 3.21T + 71T^{2} \)
73 \( 1 + 0.394T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 2.51T + 83T^{2} \)
89 \( 1 + 3.21T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−9.087462202229563370555152421886, −7.86855162062521171114478256307, −6.88429159324680615843438523452, −6.44395620868282157481517796075, −5.52491560982247755862769139269, −5.05361876185234539468791431815, −4.22725829379988543934450388810, −3.21452038377598959743896610138, −2.48005351212767141663109881742, −1.40122292485214995029027481810, 1.40122292485214995029027481810, 2.48005351212767141663109881742, 3.21452038377598959743896610138, 4.22725829379988543934450388810, 5.05361876185234539468791431815, 5.52491560982247755862769139269, 6.44395620868282157481517796075, 6.88429159324680615843438523452, 7.86855162062521171114478256307, 9.087462202229563370555152421886

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