Properties

Label 2-2394-1.1-c1-0-41
Degree $2$
Conductor $2394$
Sign $-1$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s − 4.82·11-s + 5.65·13-s + 14-s + 16-s − 6.82·17-s − 19-s − 2·20-s − 4.82·22-s − 4·23-s − 25-s + 5.65·26-s + 28-s − 2·29-s − 4.82·31-s + 32-s − 6.82·34-s − 2·35-s − 8.48·37-s − 38-s − 2·40-s + 3.65·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s − 1.45·11-s + 1.56·13-s + 0.267·14-s + 0.250·16-s − 1.65·17-s − 0.229·19-s − 0.447·20-s − 1.02·22-s − 0.834·23-s − 0.200·25-s + 1.10·26-s + 0.188·28-s − 0.371·29-s − 0.867·31-s + 0.176·32-s − 1.17·34-s − 0.338·35-s − 1.39·37-s − 0.162·38-s − 0.316·40-s + 0.571·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
19 \( 1 + T \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4.82T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 - 7.65T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 - 7.65T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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

−8.436271607752536886521661730061, −7.76344836819151572356159532501, −7.10029060411700939382893004477, −6.06063819866439465166710441605, −5.45095517266265691246580779339, −4.33133146033105419204605526577, −3.94589088025960481820071415042, −2.82478785985257348812006980904, −1.80024228504790852269968571759, 0, 1.80024228504790852269968571759, 2.82478785985257348812006980904, 3.94589088025960481820071415042, 4.33133146033105419204605526577, 5.45095517266265691246580779339, 6.06063819866439465166710441605, 7.10029060411700939382893004477, 7.76344836819151572356159532501, 8.436271607752536886521661730061

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