Properties

Label 2-1710-1.1-c1-0-21
Degree $2$
Conductor $1710$
Sign $-1$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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 − 2·7-s − 8-s − 10-s − 2·11-s − 4·13-s + 2·14-s + 16-s + 6·17-s + 19-s + 20-s + 2·22-s + 8·23-s + 25-s + 4·26-s − 2·28-s − 6·29-s − 8·31-s − 32-s − 6·34-s − 2·35-s − 8·37-s − 38-s − 40-s − 12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.603·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.223·20-s + 0.426·22-s + 1.66·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.338·35-s − 1.31·37-s − 0.162·38-s − 0.158·40-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -1)\)

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

−9.143133065217811529689328747972, −8.187387201860905084031521770453, −7.22290842794907741044182939276, −6.90693481045620505577551664629, −5.54073882913907745794997199657, −5.20990388829029218612908258783, −3.51591980576609749146844974643, −2.79103108058854080634904687342, −1.57154125662303517731277908738, 0, 1.57154125662303517731277908738, 2.79103108058854080634904687342, 3.51591980576609749146844974643, 5.20990388829029218612908258783, 5.54073882913907745794997199657, 6.90693481045620505577551664629, 7.22290842794907741044182939276, 8.187387201860905084031521770453, 9.143133065217811529689328747972

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