Properties

Label 2-8550-1.1-c1-0-122
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 − 2·7-s + 8-s + 3.46·11-s − 5.46·13-s − 2·14-s + 16-s − 3.46·17-s + 19-s + 3.46·22-s + 6.92·23-s − 5.46·26-s − 2·28-s − 3.46·29-s − 1.46·31-s + 32-s − 3.46·34-s + 1.46·37-s + 38-s − 5.46·43-s + 3.46·44-s + 6.92·46-s + 6.92·47-s − 3·49-s − 5.46·52-s − 12.9·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.755·7-s + 0.353·8-s + 1.04·11-s − 1.51·13-s − 0.534·14-s + 0.250·16-s − 0.840·17-s + 0.229·19-s + 0.738·22-s + 1.44·23-s − 1.07·26-s − 0.377·28-s − 0.643·29-s − 0.262·31-s + 0.176·32-s − 0.594·34-s + 0.240·37-s + 0.162·38-s − 0.833·43-s + 0.522·44-s + 1.02·46-s + 1.01·47-s − 0.428·49-s − 0.757·52-s − 1.77·53-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 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 1.46T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 - 4.92T + 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−7.10919783261138359579092708112, −6.80580520404008461037141790375, −6.11006927922105614832268231763, −5.22960932950077418406640748047, −4.66338194636155862444136035613, −3.89564860960273519275571740348, −3.11210517974203165509137301249, −2.44406911474888411034033916247, −1.40511420006487770205224781131, 0, 1.40511420006487770205224781131, 2.44406911474888411034033916247, 3.11210517974203165509137301249, 3.89564860960273519275571740348, 4.66338194636155862444136035613, 5.22960932950077418406640748047, 6.11006927922105614832268231763, 6.80580520404008461037141790375, 7.10919783261138359579092708112

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