Properties

Label 2-2070-1.1-c1-0-19
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 + 3.08·7-s − 8-s − 10-s + 6.46·11-s + 3.95·13-s − 3.08·14-s + 16-s + 3.43·17-s + 3.08·19-s + 20-s − 6.46·22-s + 23-s + 25-s − 3.95·26-s + 3.08·28-s − 0.863·29-s − 5.95·31-s − 32-s − 3.43·34-s + 3.08·35-s − 7.03·37-s − 3.08·38-s − 40-s − 5.60·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.16·7-s − 0.353·8-s − 0.316·10-s + 1.95·11-s + 1.09·13-s − 0.825·14-s + 0.250·16-s + 0.832·17-s + 0.708·19-s + 0.223·20-s − 1.37·22-s + 0.208·23-s + 0.200·25-s − 0.774·26-s + 0.583·28-s − 0.160·29-s − 1.06·31-s − 0.176·32-s − 0.588·34-s + 0.521·35-s − 1.15·37-s − 0.500·38-s − 0.158·40-s − 0.875·41-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: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.014855758\)
\(L(\frac12)\) \(\approx\) \(2.014855758\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.08T + 7T^{2} \)
11 \( 1 - 6.46T + 11T^{2} \)
13 \( 1 - 3.95T + 13T^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
29 \( 1 + 0.863T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 + 7.03T + 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−9.072461687750138940451419152941, −8.525291059385044318268314902813, −7.64505031620058481563203189489, −6.87976357111305687994787129829, −6.04544570229625922977554260925, −5.29149124876157713310563113285, −4.11139061853621326477343850955, −3.25554100253115218425793341542, −1.62531890434199560998945431872, −1.29526058308459081231307607058, 1.29526058308459081231307607058, 1.62531890434199560998945431872, 3.25554100253115218425793341542, 4.11139061853621326477343850955, 5.29149124876157713310563113285, 6.04544570229625922977554260925, 6.87976357111305687994787129829, 7.64505031620058481563203189489, 8.525291059385044318268314902813, 9.072461687750138940451419152941

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