Properties

Label 2-2070-1.1-c1-0-0
Degree $2$
Conductor $2070$
Sign $1$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 − 5.12·7-s − 8-s + 10-s − 5.12·11-s + 2·13-s + 5.12·14-s + 16-s − 7.12·17-s + 4·19-s − 20-s + 5.12·22-s − 23-s + 25-s − 2·26-s − 5.12·28-s − 2·29-s − 32-s + 7.12·34-s + 5.12·35-s − 7.12·37-s − 4·38-s + 40-s − 2·41-s − 5.12·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.93·7-s − 0.353·8-s + 0.316·10-s − 1.54·11-s + 0.554·13-s + 1.36·14-s + 0.250·16-s − 1.72·17-s + 0.917·19-s − 0.223·20-s + 1.09·22-s − 0.208·23-s + 0.200·25-s − 0.392·26-s − 0.968·28-s − 0.371·29-s − 0.176·32-s + 1.22·34-s + 0.865·35-s − 1.17·37-s − 0.648·38-s + 0.158·40-s − 0.312·41-s − 0.772·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4022858320\)
\(L(\frac12)\) \(\approx\) \(0.4022858320\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 + 5.12T + 7T^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 7.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 0.876T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 5.12T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 3.12T + 89T^{2} \)
97 \( 1 - 0.246T + 97T^{2} \)
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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

−9.001689272658231811437568414186, −8.577001114223972297420009252660, −7.45048294540669792161821743024, −6.96847314861357414786976546877, −6.14012706877911246905249672678, −5.31607040833313925434530722999, −3.96803155340569292580337095906, −3.11439405559670064525366173300, −2.32612613328561852707480352135, −0.43167941363320819501824277025, 0.43167941363320819501824277025, 2.32612613328561852707480352135, 3.11439405559670064525366173300, 3.96803155340569292580337095906, 5.31607040833313925434530722999, 6.14012706877911246905249672678, 6.96847314861357414786976546877, 7.45048294540669792161821743024, 8.577001114223972297420009252660, 9.001689272658231811437568414186

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