Properties

Label 2-8550-1.1-c1-0-102
Degree $2$
Conductor $8550$
Sign $-1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 2·11-s + 4·13-s + 16-s − 19-s − 2·22-s − 8·23-s − 4·26-s − 2·29-s − 2·31-s − 32-s + 8·37-s + 38-s − 2·41-s − 4·43-s + 2·44-s + 8·46-s + 4·47-s − 7·49-s + 4·52-s − 2·53-s + 2·58-s − 10·61-s + 2·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.603·11-s + 1.10·13-s + 1/4·16-s − 0.229·19-s − 0.426·22-s − 1.66·23-s − 0.784·26-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 1.31·37-s + 0.162·38-s − 0.312·41-s − 0.609·43-s + 0.301·44-s + 1.17·46-s + 0.583·47-s − 49-s + 0.554·52-s − 0.274·53-s + 0.262·58-s − 1.28·61-s + 0.254·62-s + 1/8·64-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: yes
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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.61986472172441140029633725276, −6.72870855469494060656956789272, −6.15200373065494377052148647298, −5.66587205248938053119139014089, −4.47117862991889901547772225984, −3.84290632308267884507260342246, −3.02597703266850218057670872385, −1.95286353910041906363940110984, −1.26997720328741880473546784207, 0, 1.26997720328741880473546784207, 1.95286353910041906363940110984, 3.02597703266850218057670872385, 3.84290632308267884507260342246, 4.47117862991889901547772225984, 5.66587205248938053119139014089, 6.15200373065494377052148647298, 6.72870855469494060656956789272, 7.61986472172441140029633725276

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