Properties

Label 2-9555-1.1-c1-0-178
Degree $2$
Conductor $9555$
Sign $1$
Analytic cond. $76.2970$
Root an. cond. $8.73481$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 9-s + 2·10-s + 5·11-s − 2·12-s + 13-s − 15-s − 4·16-s + 7·17-s + 2·18-s + 6·19-s + 2·20-s + 10·22-s + 3·23-s + 25-s + 2·26-s − 27-s + 2·29-s − 2·30-s − 2·31-s − 8·32-s − 5·33-s + 14·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s + 1.50·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s − 16-s + 1.69·17-s + 0.471·18-s + 1.37·19-s + 0.447·20-s + 2.13·22-s + 0.625·23-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.365·30-s − 0.359·31-s − 1.41·32-s − 0.870·33-s + 2.40·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9555\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(76.2970\)
Root analytic conductor: \(8.73481\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9555,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.104767090\)
\(L(\frac12)\) \(\approx\) \(5.104767090\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−7.33336734150810117899384110898, −6.63766955231328651971714453049, −6.22825685841877325575095618557, −5.42070052673462305634659242884, −5.12809484155897799374975575742, −4.27349695575836161639954637710, −3.42523128173611276512755811052, −3.12479027254654742175262146773, −1.71851321747435352467993608536, −0.976920050795348365188556174843, 0.976920050795348365188556174843, 1.71851321747435352467993608536, 3.12479027254654742175262146773, 3.42523128173611276512755811052, 4.27349695575836161639954637710, 5.12809484155897799374975575742, 5.42070052673462305634659242884, 6.22825685841877325575095618557, 6.63766955231328651971714453049, 7.33336734150810117899384110898

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