Properties

Label 2-8550-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4.69·7-s − 8-s − 6.40·11-s + 1.06·13-s + 4.69·14-s + 16-s − 1.91·17-s − 19-s + 6.40·22-s − 1.79·23-s − 1.06·26-s − 4.69·28-s − 2.93·29-s − 5.55·31-s − 32-s + 1.91·34-s − 11.4·37-s + 38-s + 1.14·41-s − 3.55·43-s − 6.40·44-s + 1.79·46-s − 10.8·47-s + 15.0·49-s + 1.06·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.77·7-s − 0.353·8-s − 1.93·11-s + 0.295·13-s + 1.25·14-s + 0.250·16-s − 0.465·17-s − 0.229·19-s + 1.36·22-s − 0.374·23-s − 0.208·26-s − 0.887·28-s − 0.545·29-s − 0.997·31-s − 0.176·32-s + 0.328·34-s − 1.87·37-s + 0.162·38-s + 0.178·41-s − 0.542·43-s − 0.966·44-s + 0.264·46-s − 1.58·47-s + 2.14·49-s + 0.147·52-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: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04081510999\)
\(L(\frac12)\) \(\approx\) \(0.04081510999\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.69T + 7T^{2} \)
11 \( 1 + 6.40T + 11T^{2} \)
13 \( 1 - 1.06T + 13T^{2} \)
17 \( 1 + 1.91T + 17T^{2} \)
23 \( 1 + 1.79T + 23T^{2} \)
29 \( 1 + 2.93T + 29T^{2} \)
31 \( 1 + 5.55T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 - 1.14T + 41T^{2} \)
43 \( 1 + 3.55T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 8.69T + 53T^{2} \)
59 \( 1 - 5.63T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 + 8.82T + 67T^{2} \)
71 \( 1 - 1.42T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 1.96T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.77880126047244402711177013279, −7.07982581687921454678975047221, −6.61903363701869428033691230439, −5.72810776860239521708122215781, −5.31111932536376094648547630258, −4.06680172867987293925956239617, −3.22402136845271470381309951285, −2.70042094622137294120103889590, −1.76972506590258354513283606732, −0.10201992686446687274177013857, 0.10201992686446687274177013857, 1.76972506590258354513283606732, 2.70042094622137294120103889590, 3.22402136845271470381309951285, 4.06680172867987293925956239617, 5.31111932536376094648547630258, 5.72810776860239521708122215781, 6.61903363701869428033691230439, 7.07982581687921454678975047221, 7.77880126047244402711177013279

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