Properties

Label 2-2070-1.1-c1-0-15
Degree $2$
Conductor $2070$
Sign $1$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 + 8-s + 10-s + 4·11-s − 6·13-s + 16-s + 6·17-s + 4·19-s + 20-s + 4·22-s − 23-s + 25-s − 6·26-s + 6·29-s − 8·31-s + 32-s + 6·34-s + 6·37-s + 4·38-s + 40-s − 10·41-s + 4·43-s + 4·44-s − 46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.208·23-s + 1/5·25-s − 1.17·26-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.986·37-s + 0.648·38-s + 0.158·40-s − 1.56·41-s + 0.609·43-s + 0.603·44-s − 0.147·46-s + 1.16·47-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: yes
Arithmetic: yes
Character: $\chi_{2070} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.256982269\)
\(L(\frac12)\) \(\approx\) \(3.256982269\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.382786779937371819987783736909, −8.232215862797765904742560189080, −7.29356680017341983104579693045, −6.83167002982379912689199133265, −5.71323335841138265866627321199, −5.23821231340755244483380240660, −4.23790248325532820999782331121, −3.32197498177051620753450033791, −2.36866993182259818840915365989, −1.17351616216495372278974762865, 1.17351616216495372278974762865, 2.36866993182259818840915365989, 3.32197498177051620753450033791, 4.23790248325532820999782331121, 5.23821231340755244483380240660, 5.71323335841138265866627321199, 6.83167002982379912689199133265, 7.29356680017341983104579693045, 8.232215862797765904742560189080, 9.382786779937371819987783736909

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