Properties

Label 2-8550-1.1-c1-0-129
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 0.732·7-s + 8-s − 3.73·11-s − 2.73·13-s + 0.732·14-s + 16-s + 6.19·17-s − 19-s − 3.73·22-s − 5.92·23-s − 2.73·26-s + 0.732·28-s + 1.73·29-s − 2.46·31-s + 32-s + 6.19·34-s − 2·37-s − 38-s + 2.92·41-s − 8.19·43-s − 3.73·44-s − 5.92·46-s − 3.46·47-s − 6.46·49-s − 2.73·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.276·7-s + 0.353·8-s − 1.12·11-s − 0.757·13-s + 0.195·14-s + 0.250·16-s + 1.50·17-s − 0.229·19-s − 0.795·22-s − 1.23·23-s − 0.535·26-s + 0.138·28-s + 0.321·29-s − 0.442·31-s + 0.176·32-s + 1.06·34-s − 0.328·37-s − 0.162·38-s + 0.457·41-s − 1.24·43-s − 0.562·44-s − 0.874·46-s − 0.505·47-s − 0.923·49-s − 0.378·52-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 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 - 6.19T + 17T^{2} \)
23 \( 1 + 5.92T + 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 + 2.46T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 2.92T + 41T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 1.73T + 53T^{2} \)
59 \( 1 - 8.19T + 59T^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + 0.267T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 2.46T + 73T^{2} \)
79 \( 1 + 5.53T + 79T^{2} \)
83 \( 1 + 3.73T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−7.40787614948634335750946393799, −6.76734650044120881290759287310, −5.77167778965855642121437966055, −5.39580368359957960871653916932, −4.72128811608547781108258003975, −3.90246778591016448449898593674, −3.08848644955581696429410110980, −2.38232015590899424193308792041, −1.46842671241138230954428829894, 0, 1.46842671241138230954428829894, 2.38232015590899424193308792041, 3.08848644955581696429410110980, 3.90246778591016448449898593674, 4.72128811608547781108258003975, 5.39580368359957960871653916932, 5.77167778965855642121437966055, 6.76734650044120881290759287310, 7.40787614948634335750946393799

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