Properties

Label 2-2790-1.1-c1-0-27
Degree $2$
Conductor $2790$
Sign $1$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 4·7-s + 8-s − 10-s + 4·11-s + 2·13-s + 4·14-s + 16-s − 20-s + 4·22-s − 2·23-s + 25-s + 2·26-s + 4·28-s − 2·29-s − 31-s + 32-s − 4·35-s + 2·37-s − 40-s + 4·43-s + 4·44-s − 2·46-s + 9·49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.223·20-s + 0.852·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.158·40-s + 0.609·43-s + 0.603·44-s − 0.294·46-s + 9/7·49-s + 0.141·50-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.597765796\)
\(L(\frac12)\) \(\approx\) \(3.597765796\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.654500372321901816842255950761, −7.970440067639789960903941990640, −7.31741015618283792753525367787, −6.41613721269874027079435062809, −5.66404730325203102454538368625, −4.72010516126955509032073982441, −4.17241278723967403560521862244, −3.37996514411777076461954948223, −2.04448547604731713691349330231, −1.19304226927182692012665951795, 1.19304226927182692012665951795, 2.04448547604731713691349330231, 3.37996514411777076461954948223, 4.17241278723967403560521862244, 4.72010516126955509032073982441, 5.66404730325203102454538368625, 6.41613721269874027079435062809, 7.31741015618283792753525367787, 7.970440067639789960903941990640, 8.654500372321901816842255950761

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