Properties

Label 2-8550-1.1-c1-0-139
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 + 2·7-s + 8-s − 3·11-s + 2·13-s + 2·14-s + 16-s − 6·17-s + 19-s − 3·22-s − 3·23-s + 2·26-s + 2·28-s − 3·29-s − 7·31-s + 32-s − 6·34-s + 2·37-s + 38-s + 6·41-s − 10·43-s − 3·44-s − 3·46-s − 12·47-s − 3·49-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.904·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.639·22-s − 0.625·23-s + 0.392·26-s + 0.377·28-s − 0.557·29-s − 1.25·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.162·38-s + 0.937·41-s − 1.52·43-s − 0.452·44-s − 0.442·46-s − 1.75·47-s − 3/7·49-s + 0.277·52-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: 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.38967402963890841299555741099, −6.68833380712143289688311273077, −5.93722294801387892582077561539, −5.28331966539317012419250168052, −4.66002635323681776702943061173, −3.98683350787584705334662275889, −3.14479129531348743785662948341, −2.21549742906358429236877500482, −1.57619574144439142337024860722, 0, 1.57619574144439142337024860722, 2.21549742906358429236877500482, 3.14479129531348743785662948341, 3.98683350787584705334662275889, 4.66002635323681776702943061173, 5.28331966539317012419250168052, 5.93722294801387892582077561539, 6.68833380712143289688311273077, 7.38967402963890841299555741099

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