Properties

Label 2-1710-1.1-c1-0-18
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s + 6·11-s + 4·14-s + 16-s − 4·17-s − 19-s − 20-s − 6·22-s + 4·23-s + 25-s − 4·28-s + 10·29-s − 2·31-s − 32-s + 4·34-s + 4·35-s − 4·37-s + 38-s + 40-s − 10·41-s − 12·43-s + 6·44-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 + 1.80·11-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.229·19-s − 0.223·20-s − 1.27·22-s + 0.834·23-s + 1/5·25-s − 0.755·28-s + 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.676·35-s − 0.657·37-s + 0.162·38-s + 0.158·40-s − 1.56·41-s − 1.82·43-s + 0.904·44-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: $\chi_{1710} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.800959160549418353404093061144, −8.572524300455301289677585357698, −7.02457092631358762841874739945, −6.77933183375435135203347735454, −6.12618704941329089496816400736, −4.66703219217832155170913346055, −3.65422119588025401355148482584, −2.92585686336718744421991303103, −1.42375377672046688546052263655, 0, 1.42375377672046688546052263655, 2.92585686336718744421991303103, 3.65422119588025401355148482584, 4.66703219217832155170913346055, 6.12618704941329089496816400736, 6.77933183375435135203347735454, 7.02457092631358762841874739945, 8.572524300455301289677585357698, 8.800959160549418353404093061144

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