Properties

Label 2-2790-1.1-c1-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 2.56·7-s − 8-s + 10-s − 2.56·11-s + 2·13-s − 2.56·14-s + 16-s + 3.12·17-s − 7.68·19-s − 20-s + 2.56·22-s − 1.43·23-s + 25-s − 2·26-s + 2.56·28-s − 7.12·29-s + 31-s − 32-s − 3.12·34-s − 2.56·35-s − 3.12·37-s + 7.68·38-s + 40-s − 7.12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.968·7-s − 0.353·8-s + 0.316·10-s − 0.772·11-s + 0.554·13-s − 0.684·14-s + 0.250·16-s + 0.757·17-s − 1.76·19-s − 0.223·20-s + 0.546·22-s − 0.299·23-s + 0.200·25-s − 0.392·26-s + 0.484·28-s − 1.32·29-s + 0.179·31-s − 0.176·32-s − 0.535·34-s − 0.432·35-s − 0.513·37-s + 1.24·38-s + 0.158·40-s − 1.11·41-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: \(1\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
31 \( 1 - T \)
good7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + 7.68T + 19T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + 5.12T + 47T^{2} \)
53 \( 1 + 7.43T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 - 7.68T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 4.31T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.380274100276154634723532407800, −7.84840495452785283897936671622, −7.18510630725992642247790849893, −6.18348433413767279723290568307, −5.40386405569995383148600131527, −4.45665963167379737437363568466, −3.56624755388065188473922082707, −2.37587517893091441460462867653, −1.46553348244527503199022869130, 0, 1.46553348244527503199022869130, 2.37587517893091441460462867653, 3.56624755388065188473922082707, 4.45665963167379737437363568466, 5.40386405569995383148600131527, 6.18348433413767279723290568307, 7.18510630725992642247790849893, 7.84840495452785283897936671622, 8.380274100276154634723532407800

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