Properties

Label 2-8550-1.1-c1-0-130
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 + 0.761·7-s + 8-s + 0.864·11-s − 5.62·13-s + 0.761·14-s + 16-s − 3.62·17-s + 19-s + 0.864·22-s − 8.01·23-s − 5.62·26-s + 0.761·28-s + 7.35·29-s + 8.11·31-s + 32-s − 3.62·34-s − 0.476·37-s + 38-s + 2.65·41-s − 6.86·43-s + 0.864·44-s − 8.01·46-s + 1.25·47-s − 6.42·49-s − 5.62·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.287·7-s + 0.353·8-s + 0.260·11-s − 1.56·13-s + 0.203·14-s + 0.250·16-s − 0.879·17-s + 0.229·19-s + 0.184·22-s − 1.67·23-s − 1.10·26-s + 0.143·28-s + 1.36·29-s + 1.45·31-s + 0.176·32-s − 0.621·34-s − 0.0783·37-s + 0.162·38-s + 0.415·41-s − 1.04·43-s + 0.130·44-s − 1.18·46-s + 0.182·47-s − 0.917·49-s − 0.780·52-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 \( 1 - 0.761T + 7T^{2} \)
11 \( 1 - 0.864T + 11T^{2} \)
13 \( 1 + 5.62T + 13T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
23 \( 1 + 8.01T + 23T^{2} \)
29 \( 1 - 7.35T + 29T^{2} \)
31 \( 1 - 8.11T + 31T^{2} \)
37 \( 1 + 0.476T + 37T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 + 6.86T + 43T^{2} \)
47 \( 1 - 1.25T + 47T^{2} \)
53 \( 1 - 2.37T + 53T^{2} \)
59 \( 1 + 4.49T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 1.03T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 0.270T + 83T^{2} \)
89 \( 1 + 0.387T + 89T^{2} \)
97 \( 1 - 8.50T + 97T^{2} \)
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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

−7.38252061590118875795382936914, −6.58321268656818751121190727066, −6.14963542833399030206752401885, −5.16938426109606221631984888405, −4.59763757447936182882148379644, −4.12601504695656701092241953371, −2.97231525182228017746446668479, −2.41060726606394308702123904767, −1.47894176251802613675484926173, 0, 1.47894176251802613675484926173, 2.41060726606394308702123904767, 2.97231525182228017746446668479, 4.12601504695656701092241953371, 4.59763757447936182882148379644, 5.16938426109606221631984888405, 6.14963542833399030206752401885, 6.58321268656818751121190727066, 7.38252061590118875795382936914

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