Properties

Label 2-2790-1.1-c1-0-33
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 + 1.07·7-s − 8-s + 10-s − 6.04·11-s − 0.290·13-s − 1.07·14-s + 16-s − 1.07·17-s + 5.26·19-s − 20-s + 6.04·22-s + 4.34·23-s + 25-s + 0.290·26-s + 1.07·28-s + 9.31·29-s + 31-s − 32-s + 1.07·34-s − 1.07·35-s − 2.44·37-s − 5.26·38-s + 40-s − 5.60·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.407·7-s − 0.353·8-s + 0.316·10-s − 1.82·11-s − 0.0806·13-s − 0.288·14-s + 0.250·16-s − 0.261·17-s + 1.20·19-s − 0.223·20-s + 1.28·22-s + 0.904·23-s + 0.200·25-s + 0.0570·26-s + 0.203·28-s + 1.72·29-s + 0.179·31-s − 0.176·32-s + 0.184·34-s − 0.182·35-s − 0.402·37-s − 0.853·38-s + 0.158·40-s − 0.874·41-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 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 + 6.04T + 11T^{2} \)
13 \( 1 + 0.290T + 13T^{2} \)
17 \( 1 + 1.07T + 17T^{2} \)
19 \( 1 - 5.26T + 19T^{2} \)
23 \( 1 - 4.34T + 23T^{2} \)
29 \( 1 - 9.31T + 29T^{2} \)
37 \( 1 + 2.44T + 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 - 1.75T + 47T^{2} \)
53 \( 1 - 6.44T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 4.68T + 71T^{2} \)
73 \( 1 + 8.18T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 3.55T + 83T^{2} \)
89 \( 1 + 3.84T + 89T^{2} \)
97 \( 1 - 2.49T + 97T^{2} \)
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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

−8.352588274085450218443839301328, −7.75610669420067245234949282626, −7.21529294736749966187554232683, −6.27818910681276494194686223980, −5.14889988964788260392122932379, −4.77203662143397282599381711181, −3.24609115963682442183669842703, −2.66437745415036059099628702332, −1.33947502059879646237763624997, 0, 1.33947502059879646237763624997, 2.66437745415036059099628702332, 3.24609115963682442183669842703, 4.77203662143397282599381711181, 5.14889988964788260392122932379, 6.27818910681276494194686223980, 7.21529294736749966187554232683, 7.75610669420067245234949282626, 8.352588274085450218443839301328

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