Properties

Label 2-8550-1.1-c1-0-140
Degree $2$
Conductor $8550$
Sign $-1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
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 + 4.42·7-s + 8-s − 2.62·11-s − 5.80·13-s + 4.42·14-s + 16-s − 3.80·17-s + 19-s − 2.62·22-s − 2.62·23-s − 5.80·26-s + 4.42·28-s − 3.37·29-s − 4.42·31-s + 32-s − 3.80·34-s − 5.80·37-s + 38-s − 5.67·41-s + 10.9·43-s − 2.62·44-s − 2.62·46-s + 2.62·47-s + 12.6·49-s − 5.80·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.67·7-s + 0.353·8-s − 0.790·11-s − 1.61·13-s + 1.18·14-s + 0.250·16-s − 0.923·17-s + 0.229·19-s − 0.559·22-s − 0.546·23-s − 1.13·26-s + 0.836·28-s − 0.627·29-s − 0.795·31-s + 0.176·32-s − 0.652·34-s − 0.954·37-s + 0.162·38-s − 0.885·41-s + 1.67·43-s − 0.395·44-s − 0.386·46-s + 0.382·47-s + 1.80·49-s − 0.805·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(68.2720\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 - 4.42T + 7T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 + 5.80T + 13T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + 3.37T + 29T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 + 5.80T + 37T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 2.62T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 - 4.75T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 4.42T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 7.37T + 97T^{2} \)
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

−7.47328931425999704860138491765, −6.89822281176198746450474823625, −5.68941102478384567301592458254, −5.33616296523912045398659182247, −4.59003287182386651509907057275, −4.22268787188847659401012501087, −2.98159096782937508208356549140, −2.21215486267884727371759061022, −1.63857972936392363372430223311, 0, 1.63857972936392363372430223311, 2.21215486267884727371759061022, 2.98159096782937508208356549140, 4.22268787188847659401012501087, 4.59003287182386651509907057275, 5.33616296523912045398659182247, 5.68941102478384567301592458254, 6.89822281176198746450474823625, 7.47328931425999704860138491765

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