Properties

Label 2-2070-1.1-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 4·14-s + 16-s − 4·17-s + 8·19-s + 20-s + 23-s + 25-s − 4·28-s + 2·29-s + 4·31-s − 32-s + 4·34-s − 4·35-s − 2·37-s − 8·38-s − 40-s − 8·41-s − 4·43-s − 46-s − 8·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.755·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.676·35-s − 0.328·37-s − 1.29·38-s − 0.158·40-s − 1.24·41-s − 0.609·43-s − 0.147·46-s − 1.16·47-s + 9/7·49-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.933171196373164436028073628791, −8.047403151846088508381664903301, −7.02078577529983508395697282954, −6.58094930913824246215386749564, −5.78281516922020646818108599784, −4.77008429136827408025826825802, −3.35966149519111135869782665267, −2.82354330440148796947542327122, −1.45402989302954842587531760308, 0, 1.45402989302954842587531760308, 2.82354330440148796947542327122, 3.35966149519111135869782665267, 4.77008429136827408025826825802, 5.78281516922020646818108599784, 6.58094930913824246215386749564, 7.02078577529983508395697282954, 8.047403151846088508381664903301, 8.933171196373164436028073628791

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