Properties

Label 2-8550-1.1-c1-0-142
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 + 3.86·7-s + 8-s − 1.03·11-s − 3.86·13-s + 3.86·14-s + 16-s − 0.535·17-s − 19-s − 1.03·22-s − 5.46·23-s − 3.86·26-s + 3.86·28-s − 4.62·29-s − 10.9·31-s + 32-s − 0.535·34-s − 1.79·37-s − 38-s − 1.79·41-s − 6.69·43-s − 1.03·44-s − 5.46·46-s − 2.53·47-s + 7.92·49-s − 3.86·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.46·7-s + 0.353·8-s − 0.312·11-s − 1.07·13-s + 1.03·14-s + 0.250·16-s − 0.129·17-s − 0.229·19-s − 0.220·22-s − 1.13·23-s − 0.757·26-s + 0.730·28-s − 0.858·29-s − 1.96·31-s + 0.176·32-s − 0.0919·34-s − 0.294·37-s − 0.162·38-s − 0.280·41-s − 1.02·43-s − 0.156·44-s − 0.805·46-s − 0.369·47-s + 1.13·49-s − 0.535·52-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 \( 1 - 3.86T + 7T^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
13 \( 1 + 3.86T + 13T^{2} \)
17 \( 1 + 0.535T + 17T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + 4.62T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + 1.79T + 37T^{2} \)
41 \( 1 + 1.79T + 41T^{2} \)
43 \( 1 + 6.69T + 43T^{2} \)
47 \( 1 + 2.53T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6.96T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 4.14T + 71T^{2} \)
73 \( 1 + 3.58T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 - 3.86T + 89T^{2} \)
97 \( 1 - 2.55T + 97T^{2} \)
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

−7.50269607040654427646491681912, −6.79202021815565730933913396940, −5.80742519492580437630431964127, −5.26160731440865400173229999369, −4.70492343887779334889799690128, −4.03275512427787268705003381220, −3.15326982473643529882282292414, −2.03110620322102906332515267590, −1.73675145047941810413858599166, 0, 1.73675145047941810413858599166, 2.03110620322102906332515267590, 3.15326982473643529882282292414, 4.03275512427787268705003381220, 4.70492343887779334889799690128, 5.26160731440865400173229999369, 5.80742519492580437630431964127, 6.79202021815565730933913396940, 7.50269607040654427646491681912

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