Properties

Label 2-2790-1.1-c1-0-24
Degree $2$
Conductor $2790$
Sign $1$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
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-s + 4-s + 5-s + 3.53·7-s − 8-s − 10-s + 1.53·11-s + 6·13-s − 3.53·14-s + 16-s + 4·17-s − 3.53·19-s + 20-s − 1.53·22-s + 1.53·23-s + 25-s − 6·26-s + 3.53·28-s + 31-s − 32-s − 4·34-s + 3.53·35-s + 9.06·37-s + 3.53·38-s − 40-s + 9.06·41-s − 0.468·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.33·7-s − 0.353·8-s − 0.316·10-s + 0.461·11-s + 1.66·13-s − 0.943·14-s + 0.250·16-s + 0.970·17-s − 0.810·19-s + 0.223·20-s − 0.326·22-s + 0.319·23-s + 0.200·25-s − 1.17·26-s + 0.667·28-s + 0.179·31-s − 0.176·32-s − 0.685·34-s + 0.596·35-s + 1.48·37-s + 0.572·38-s − 0.158·40-s + 1.41·41-s − 0.0715·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(22.2782\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.027118063\)
\(L(\frac12)\) \(\approx\) \(2.027118063\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
31 \( 1 - T \)
good7 \( 1 - 3.53T + 7T^{2} \)
11 \( 1 - 1.53T + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 3.53T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 - 9.06T + 37T^{2} \)
41 \( 1 - 9.06T + 41T^{2} \)
43 \( 1 + 0.468T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 5.53T + 53T^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 4.46T + 71T^{2} \)
73 \( 1 + 0.468T + 73T^{2} \)
79 \( 1 + 0.468T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + 1.53T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−8.805598378178777518083228619532, −8.051302926031878531550217920613, −7.64058531780816519101915685917, −6.36081126922860334700090049837, −6.02826560522312880660647924233, −4.93981276658805860548767706815, −4.05651659749537333518747540161, −2.95168776743297228734946623259, −1.69779465124660635776341477032, −1.13112750137493544231039460021, 1.13112750137493544231039460021, 1.69779465124660635776341477032, 2.95168776743297228734946623259, 4.05651659749537333518747540161, 4.93981276658805860548767706815, 6.02826560522312880660647924233, 6.36081126922860334700090049837, 7.64058531780816519101915685917, 8.051302926031878531550217920613, 8.805598378178777518083228619532

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